Pretty sure there's a term for all of these things, but I don't know whether it's "reading practices" or something totally different. But here's some brief info anyway, half of which is leeched off my classmates' powerpoints.
Information
Marxist criticism: Marxism is a way of seeing the world through economic terms. For example, Marxists see history as a series of class struggles between whatever the upper and lower classes of the period happen to be, whether they be slaves and masters or workers and capitalists. Marxists believe that systems of production of goods exist until they are no longer sufficient and are replaced by a new system. They are often critical of capitalism as they believe that, while the idea that capitalists have to produce what people want to maintain profits might mean that resources are distributed well, it can also go the other way, because it could mean that capitalists are then producing more for people who can afford it to the detriment of those who cannot, as well as to the detriment of the environment. There's a whole bunch of other ideas that are related to Marxism, which you can read about at http://socialsciences.arts.unsw.edu.au/tsw/Marx.html.
Psychoanalytic criticism: This is mainly to do with Freud's ideas about how the mind works. Freud believed that the mind could be divided up into the id, the ego and the superego. Now, I'm not sure what the ego did, but what I do remember is that the id is basically the unconscious desires and the superego is the mind's means of regulating these desires. A significant number of these desires, according to Freud, have something or other to do with repressed sexuality. I mean, he even divides our life into the oral, anal, phallic and genitals stages. If you don't believe me, go look up the case about Freud and Dora.
New Historicist criticism: I think this is mainly to do with how attitudes at the time influenced a text, as opposed to just looking at context which is where the concrete events themselves influence the text? I'm not entirely sure, and I'm probably completely wrong here. Maybe you can just put down the stuff that I wrote on my Context page after all.
There's other ways of reading, like Feminist criticism, but I've been procrastinating all day and it's now 9pm so I'd better hurry up and get this over with. (I swear I started writing the first article at about 9am this morning... shows how much I procrastinate.)
Reading Prufrock Through These Lenses
Marxist Criticism: My powerpoint was "Prufrock can be read as an exploration of the Marxist idea that the relentless pursuit of profits in a capitalist society advantages the upper classes while disadvantaging lower classes and the environment through poetic devices such as imagery, similes and intertextuality." On my first slide, I wrote about the representations of the advantaged upper classes. There are consistent references to upper-class goods and lifestyle, such as the "taking of a toast and tea," "tea and cakes and ices" and "porcelain." There are also references to the aesthetic qualities present in upper-class life, such as "arms that are braceleted" and people who "prepare a face" to meet each other. Also, the word "time" is used constantly, giving the sense that the upper classes have plenty of leisure time to spare- and who doesn't like leisure time? There's "time for you and time for me," before they even get to have some delicious upper-class tea! What more could you want? Oh, and there's also that saying that "time is money" as well, which only serves to make the upper-classes look even wealthier. Brilliant.
The disadvantaged lower classes are shown through the imagery in the poem as well as through the juxtaposition between the upper- and lower-classes. "Cheap hotels" and "sawdust restaurants" give the impression of poverty and dirtiness, the aural imagery of "the muttering retreats of restless nights" make the city seem unpleasant and unsafe (see my post on Context) and the visual/ kinesthetic imagery of the yellow cat/smoke makes the streets seem forbidding and almost poisonous due to the way that it has been described as "yellow fog" and "yellow smoke" rubbing itself against window panes. I mean, you have poisonous gases rubbing themselves up against your house, for goodness' sake. Isn't that a bit too close for comfort? The "lonely men in shirt sleeves" mentioned later in the poem can be read as a representation of disconnected workers in capitalist society. All of this is juxtaposed against the "tea and cakes and ices" and porcelain of the upper-classes, a stark contrast which is accentuated by the constant swapping between the two scenarios.
The damaged environment is shown through kinaesthetic imagery, similes and visual imagery. In the kinaesthetic imagery category, we have "let fall upon its back the soot that falls from chimneys," "the yellow fog that rubs its back" and "the yellow smoke that rubs its muzzle." The pollution's EVERYWHERE, rubbing against us and everything, and that ain't pretty. And then there's my favourite simile, "when the evening is spread out against the sky like a patient etherised upon a table." Etherisation is an old anaesthetisation technique in which the patient cannot move but still has his or her mental faculties intact. Comparing the sky to an etherised patient makes it seem as though we have made nature helpless. Finally, the visual imagery of "where wind blows the water white and black" possibly shows the effects of pollution: why is the water black? Is it because of the pollution from the factories?
Freudian criticism: Finding stuff to say about this one is fun because it means you get to look for innuendos. On the other hand, it might not be fun for the same reason.
At the moment I'm just looking at two of my classmates' powerpoints. One of them has split up her analysis into symbols of desire, symbols of repression and symbols of impotence. Another group has split up their analysis into representations of the id (which I guess could correspond to "desire" in the other powerpoint), the superego (which I guess could correspond to "repression" and "impotence" in the other powerpoint) and where the two come to a head in what they term "the battle."
I remember hearing somewhere that you can copy 10% of someone else's work without it being deemed plagiarism, so that's pretty much what I'm going to do. (Actually the real reason why I'm being brief here is not because I care about plagiarism but because I'm too lazy to write a lot.) So here goes:
Okay. Symbols of desire: "Do I dare to eat a peach?" His wanting to symbolises desire, but his questioning about it symbolises his restraint (superego). The mermaids are a reference to the sirens in Homer's Odyssey. Look it up because I'm too lazy to explain. (Okay, I might write it in some other time, but not now.) Arms- symbolises his desire for women. Once in class we were talking about how referring to people as "arms" is an example of synechdoche, in which a part represents the whole (guess I could have talked about that in my "fragmentation" part of my post on Context). Wow, that was a random point that doesn't really fit here. Anyway, perfume is another symbol of temptation and desire.
Symbols of repression: "Oh, do not ask, 'What is it?'"- never says his overwhelming question because something is stopping him. Damn. Then there's the "preparing a face" to almost construct a false identity and the prudent and respectful characteristics of the "attendant lord" which this classmate of mine reckons is Prufrock's excuse for not showing strong emotion.
Symbols of impotence: The mermaids- "I do not think that they will sing to me." Not knowing whether he will "have the strength to force the moment to it crisis." References to growing old symbolise his physical and sexual impotence. Then there's "restless nights in one-night cheap hotels," which I guess symbolises impotence if you associate "one-night cheap hotels" with that kind of behaviour because if you do then "restless nights" sort of symbolises an inability to do that kind of thing.
Okay, that's enough from me. I'm tired and I can't be bothered doing anything else. Good night to you all.
Sunday, April 14, 2013
T. S. Eliot- The Love Song of J. Alfred Prufrock- Imagery
This is my second post about this oh-so-annoying-to-analyse-for-Lit poem. My first post is on context: http://year11misadventures.blogspot.com.au/2013/04/ts-eliot-love-song-of-j-alfred-prufrock.html.
Imagery: Information
I'm not really sure what the correct definition for "imagery" is but I guess it's just words that build up really strong mental images of stuff. There are different kinds of imagery that appeal to (not sure if that's the right word) different senses. Some images may not fit neatly into one category, but into multiple categories.
Visual imagery: Imagery related to sight.
Olfactory imagery: Imagery related to smell.
Kinesthetic imagery: Imagery related to movement.
Tactile imagery: Imagery related to touch.
Aural imagery: Imagery related to sound.
Gustatory imagery: Imagery related to taste.
There's probably other types but those are the categories that I know of.
Imagery in Prufrock
I'll do visual imagery last because much more imagery fits into that category than any other category.
Kinesthetic imagery: There's the yellow smoke/fog that "[rubs] its back," that "[curls] about the house" and "slides along the street." There's also a bit where Prufrock says that he is "pinned and wriggling on the wall."
Aural imagery: There's the "muttering retreats," the "music from a farther room," and the singing mermaids.
Visual imagery: The "yellow smoke" and "yellow fog," the overall image/metaphor/whatever of the cat or whatever domestic animal that is in that stanza, "lift and drop a question on your plate," Prufrock's description of himself, "arms that are braceleted and white and bare," "water white and black" and so on.
Now, where do we go with this imagery? Well, imagery's one of those generic conventions that you can use to help back up your statements in an essay. Also, you might have a question asking how imagery achieves desired effects, and then you can ramble on about the imagery above. If you don't know what to say, just pick an image and make stuff up. Okay, maybe you shouldn't do that if you want good marks. If you're really stuck though, it's better than writing nothing at all. You'd probably get something for at least writing down what type of imagery you're referring to, though I'm not sure because I'm not the one marking these things.
Imagery: Information
I'm not really sure what the correct definition for "imagery" is but I guess it's just words that build up really strong mental images of stuff. There are different kinds of imagery that appeal to (not sure if that's the right word) different senses. Some images may not fit neatly into one category, but into multiple categories.
Visual imagery: Imagery related to sight.
Olfactory imagery: Imagery related to smell.
Kinesthetic imagery: Imagery related to movement.
Tactile imagery: Imagery related to touch.
Aural imagery: Imagery related to sound.
Gustatory imagery: Imagery related to taste.
There's probably other types but those are the categories that I know of.
Imagery in Prufrock
I'll do visual imagery last because much more imagery fits into that category than any other category.
Kinesthetic imagery: There's the yellow smoke/fog that "[rubs] its back," that "[curls] about the house" and "slides along the street." There's also a bit where Prufrock says that he is "pinned and wriggling on the wall."
Aural imagery: There's the "muttering retreats," the "music from a farther room," and the singing mermaids.
Visual imagery: The "yellow smoke" and "yellow fog," the overall image/metaphor/whatever of the cat or whatever domestic animal that is in that stanza, "lift and drop a question on your plate," Prufrock's description of himself, "arms that are braceleted and white and bare," "water white and black" and so on.
Now, where do we go with this imagery? Well, imagery's one of those generic conventions that you can use to help back up your statements in an essay. Also, you might have a question asking how imagery achieves desired effects, and then you can ramble on about the imagery above. If you don't know what to say, just pick an image and make stuff up. Okay, maybe you shouldn't do that if you want good marks. If you're really stuck though, it's better than writing nothing at all. You'd probably get something for at least writing down what type of imagery you're referring to, though I'm not sure because I'm not the one marking these things.
T.S. Eliot- The Love Song of J. Alfred Prufrock- Context
Right, so I have a Lit essay tomorrow that I'm meant to study for, but I really can't be stuffed writing practise paragraphs or essays or anything so I figure writing a blog post about Eliot instead might suffice- at the very least, it's better than just sitting there with a blank page and procrastinating.
The essay tomorrow's on Eliot's poetry, more specifically the two poems we've studied in class: "The Love Song of J. Alfred Prufrock" and "Preludes," but chances are, I'm probably going to write more on Prufrock because a) it's longer and therefore has more material to work with and b) we've discussed it more in class. We've also been told that the 3-4 questions can be on lit techniques and devices, context, genre and generic conventions, language, discourse and ideology or reading strategies. I'm going to write some notes on some of these things, as normally the bit I find hardest is planning my essay. Once that's done, the writing normally takes care of itself.
Most of the stuff here is probably from class discussions, including from other people's notes (they were put on the J drive to be shared between us...). Yeah, it's plagiarism, I know, but a) I gave them my notes on Marxism and b) I'm not passing any of this stuff off as my own, I'm just putting it all here as a means to organise my thoughts. But if you really feel that strongly, I can get rid of some of the stuff here.
Context: Information
The socio-historical context: "The Love Song of J. Alfred Prufrock" was written in the early-ish 1900s (according to Wikipedia, Eliot started writing it in 1910, but it was published in 1915). This is a time of rapid change and turmoil- I mean, WWI happened at this time! Suddenly the whole world was plunged into a war, the likes of which had never been seen before, since the advent of new, powerful weapons as well as the usage of old, outdated tactics (i.e. charging in a straight line, which would have been useful for knights on horseback but not for modern soldiers in the face of machine guns) meant that loads of people were dying in the trenches pretty quickly. Also, as well as modern weaponry, other new inventions were changing the world: communications were improving via phones and TV, and transport was also improving.
Other trends present at this time were urbanisation and secularisation, though both of these kind of continue from previous times. During the Industrial Revolution of around 1760-1820, as well as afterwards, heaps of people began moving to the cities to find work, leading to crowded and busy cities which were sometimes dirty and polluted as greener technology had not yet been invented. The busy life in these crowded cities led to people becoming more disconnected from nature, and maybe even from each other too, despite the many advances in communication technology. The increase in secularisation at this time is simply yet another stage of a gradual increase of secularisation since the Renaissance period, but this time it was largely fuelled by all of the death and devastation caused by the World War which led people to challenge their faith.
T.S. Eliot: Okay, I must admit that I know hardly anything about this guy apart from the fact that he didn't discriminate; he hated pretty much everybody. But I do have a handout here about Eliot. Let's see what it says.
It says here that T.S. Eliot was born in St Louis (a city in Missouri, USA) in 1888, but his parents originally came from New England, which, believe it or not, is also a city in the USA (sort of shows just how overly patriotic those Brits were when they colonised half the world). It sort of ties in to how my Lit teacher was saying that Eliot is American but he likes pretending that he's British.
Anyway, Eliot grew up, discovered that he liked poetry, attended Harvard and studied some high-brow works by people like Omar Khayyam, Charles Baudelaire and Jules Laforgue. He also spent some time wandering around "the poorer and seedier parts of Boston" (yes, that's what the handout says). Apparently Charles Baudelaire is sometimes considered the father of Modernism, and from him, Eliot learned that he could write poetry on rather un-poetic material like industrial cities. From Jules Laforgue, a follower of Baudelaire, he learned about the French symbolist movement in poetry, as well as other random stuff like interior monologues which can be dialogues between rival aspects of the self, ironic voice, ironic mockery of society and self, irregularly placed rhymes, abrupt changes of setting, and more.
Your own context: ...
Context: How Prufrock is shaped by context
Socio-historical context: Industrialisation and its effects are seen in the way that the city has been portrayed as dilapidated, dirty and polluted as well as being pretty damn unwelcoming through word meanings and imagery. The streets are "muttering retreats," the word "muttering" suggesting discomfort because it's as if though there is some secret that you're not being let in on, and nobody likes not being let in on a secret. I guess you can also say that the muttering and the idea that there's something that you're not being let in on also hints at the idea of disconnection between humans. Further adding to this is the "insidious intent," that word "insidious" hinting that there really is something suspicious and unpleasant going on. The nights are also "restless" and uncomfortable, and the portrayal of the city with its "one-night cheap hotels" and "sawdust restaurants" (restaurants where the ground is covered with sawdust and all your rubbish is just dumped on the floor to be swept up with the sawdust at the end of the day) makes the city seem dirty and uncared for. And, of course, there's the imagery "yellow fog" and "yellow smoke," which hints at pollution and disease. Yellow is normally a happy colour, but when related to fog and smoke, it sounds almost diseased. Likewise, the smoke "rubs its back" and "rubs its muzzle" (kinaesthetic imagery! Yay, big words) like a cat or some other domestic creature, but unlike cats and dogs, when these descriptions are used to describe smoke, it makes the smoke seem almost too close for comfort. Oh, and this is kind of going back to what I said before about disconnection but there's also some other line about "lonely men in shirt sleeves," and that very word "lonely" creates a sense of disconnection, because isn't loneliness pretty much a lack of connection with other humans?
Fragmentation is another common theme of this time period because everything, including people's ideas, are all challenged to the limits with all the crazy stuff that's happening, like the World War, and, sure enough, there's a hell of a lot of fragmentation in this poem. First of all, there's all the crazy conflicting imagery, like "when the evening is spread out against the sky like a patient etherised upon a table." (Apparently there's a fancy word for abrupt transitions in style- "bathos." Unfortunately, I don't really know how to use it in a sentence.) I mean, look at that line (or rather, two lines). Just look at it. "When the evening is spread out upon the sky" sounds all Romantic-esque with its emphasis on nature and evening (which is when 90% of romantic scenes take place, I swear), but then it abruptly turns to that poor patient. The word "patient" suggests sickness and pain- a big difference from the romance of the evening and the sky. There's also sudden transitions in settings- one line Prufrock is "at dusk through narrow streets" and three lines later he is talking about how he "should have been a pair of ragged claws scuttling across the floors of silent seas." Even the word "ragged" sounds fragmented and disjointed in itself since it does mean "uneven" or "jagged." The structure of the poem is also pretty fragmented with its breaks (I'm not sure how to describe them- the dots or whatever that divides up sections of the poem).
Hm what else? I came up with some other idea when I was in the shower, but I've forgotten it now. Damn. Guess I'll have to revisit this part.
(10 minutes later...) I've remembered now. One thing that I didn't put in the information section above is that there were still pretty big divisions between the upper- and lower-classes, even during the World War. I remember reading some Horrible Histories book that said something about how the upper classes were complaining that they had to downgrade to having parlourmaids, while soldiers were being killed by the thousands in the trenches! Such class divisions are also pretty apparent in the poem- while the city, home of the working classes, is depicted as being dirty, unpleasant and possibly unsafe, the upper class life is much different. They have "tea and cakes and ices" while the lower classes have to make do with their "sawdust restaurants," and they have "sunsets and dooryards and sprinkled streets" while the lower classes have "streets that follow like a tedious argument." The juxtaposition between the two classes is also amplified by the constant switching between lower-class scenery and upper-class scenery.
T.S. Eliot's context: Okay, first of all, everything that my handout said that Eliot learned from reading Jules Laforgue can be found in this poem. "Interior monologues which are really ironic dialogues between rival aspects of the self" are found EVERYWHERE. In fact, you could almost argue that the entire poem is an "interior monologue which [is] really [an] ironic dialogue between rival aspects of the self." We can see this in all of his incessant questioning: "Do I dare?" "Do I dare disturb the universe?" "How should I presume?" "Should I then presume?" Most of these questions are probably directed towards himself, because normally "Do I dare?" is a question that you ask yourself, not other people. Also, the use of that stanza of Dante's Inferno at the beginning of the poem, which basically means "I can tell you my story only because I know you'll never return" suggests that this poem and its questions are not to be heard by anyone, and if they are to be heard, then only to a select audience chosen with care. There's also the bit about "time yet for a hundred indecisions, and for a hundred visions and revisions" which also creates an air of uncertainty. The very word "indecision" speaks for itself- there's not enough certainty to make a decision, and any decisions then have to face "revisions." Further examples of Prufrock's divided self appear in lines 39-45, where Prufrock initially makes a decision to "turn back and descend the stair" but then worries that other people will comment on "how his hair is growing thin," and then describes his attire as "rich and modest" before worrying that other people will comment on "how his arms and legs are thin." Here Prufrock is divided by his own judgement: one aspect of himself believing that he is prepared and ready to go, and the other rival aspect of himself only able to see his faults.
I don't really have too much to say on the next three dot points here- "air of worldly fatigue," "wry, ironic voice" and "ironic mockery of society and self." Now, the reason I don't have much to say is because there is nothing to say, because I know that there's lots to say, but because I don't know how to say it. I sort of get the "worldly fatigue" and "wry, ironic voice" in this poem, but not quite enough to provide solid examples. I guess the "wry, ironic voice" and "ironic mockery" might be in all that conflicting imagery with the evening being a "patient etherised on a table," as well as in some of the other random lines like "I have measured out my life with coffee spoons" (I dunno, that just felt "wry and ironic" to me), but I'm not sure how to explain how any of that stuff contributes to a "wry, ironic voice." In fact, I'm not sure if I really know the word "ironic" all too well. I mean, I know what the word means, but I don't know what it means... okay, maybe I should stop, I'm probably just confusing you. As for the "worldly fatigue," I guess that comes in when Prufrock asks if stuff "would be worth it" and talking about himself growing old. Maybe. I'm not sure.
The next two dot points here are "vers libre: reflecting the free flow of human consciousness as it attempts to come to terms with a complex reality" and "occasional, irregularly placed rhymes." Well, I can give examples for "occasional, irregularly placed rhymes." The first two lines rhyme, then the third line has no rhyme, then the next two lines rhyme, then "hotels" and "shells" sort of rhymes, then the next two lines rhyme, then there's a line which doesn't rhyme with any other line, then another two rhyming lines. Then there's that two line couplet thingy that rhymes. Likewise, there's some rhyming lines in the next stanza, but there's also some lines which inexplicably do not rhyme, like "licked its tongue into the corners of the evening," "and seeing that it was a soft October night," and so on. As for the vers libre thing, well, I'm not too sure on what the exact literary definition of "vers libre" is but I can see that this poem does appear to be a bit more "free" (I'm pretty sure that's what "libre" means) from most of the traditional rules of poetry and it is a reflection of the human consciousness. It leaps from topic to topic, a bit like how we can move from one train of thought to another (or maybe that's just me). It compares stuff to other random stuff ("patient etherised upon a table" anyone? Oh, wait, I've used that example about 3 times already), which I do too, and although I could be unique in that, I doubt it. It also has irregular line lengths, no fixed meter, irregular rhymes and random breaks (those dividing thingos- as I've said before I don't know what to call them...), because the human consciousness doesn't have a fixed meter, regular rhyme patterns and regular line lengths, not as far as I know anyway.
Next dot point- "abrupt changes of setting, timeframe and voice." Hm. I wonder where we can find those things in Prufrock? Here's a hint: everywhere. Abrupt changes of setting? "Streets that follow like a tedious argument" vs. "In the room the women come and go, talking of Michelangelo." "Gone at dusk through narrow streets" vs. "Scuttling across the floors of silent seas." Abrupt changes of timeframe? Not so obvious, but there is that bit that Prufrock randomly starts saying "I grow old... I grow old," and I doubt that he was old before because in the next line he says "I shall wear the bottom of my trousers rolled," the word "shall" indicating that he hasn't grown old yet. Abrupt changes in voice? Should I pull out my favourite quote again? Nah, you're probably all sick of it by now.
Now for the dot point that sprouts several dot points: "Laforgian symbolism." First up is "A kind of poetry which would be psychology in the form of a dream... with flowers and scents and wind... complex symphonies with certain phrases (motifs) returning from time to time." Now, that language is too flowery and poetic for my puny brain, but I do understand the bit about certain phrases returning from time to time. "The Love Song of J. Alfred Prufrock" has some of these, such as "In the room the women come and go/ Talking of Michelangelo," "Do I dare? and "And would it have been worth it, after all." Now, we could dissect these further by talking about the effect that this repetition has on our readings of the poem and whatnot, but I can't be bothered doing that now. The next dot-point-within-a-dot-point is "Images which are absolutely precise in physical terms but endlessly suggestive in their meanings," which I think is just like that patient... You probably know that quote by now, even if you've never read the poem before. There's also the "pinned and wriggling on the wall" thing, which is pretty precise, but it can also mean other stuff like being trapped. Trapped by what? That's up to you to decide. "Building up of patterns of meaning through juxtaposition and accumulation of images," the third dot point, is seen through the juxtaposition between the upper- and lower- classes' descriptions, as I've said before. Finally, there's "contrast of sublime and banal images." I originally thought that "banal" meant something along the lines of "gruesome" or "unpleasant," like that quote that I've repeated oh-so-many times, but no, apparently it means "so lacking in originality as to be obvious and boring." I can't really say that much of the imagery in this poem is "lacking in originality," but I guess there is the "magic lantern [throwing] the nerves in patterns on a screen" followed by the more mundane "settling a pillow or throwing off a shawl."
That's all the dot points for that one covered now. Phew!
Now, I was going to cover multiple things in this post, but this post has been going on for quite a while now so I'm going to talk about this poem in multiple blog posts. Stay tuned...
The essay tomorrow's on Eliot's poetry, more specifically the two poems we've studied in class: "The Love Song of J. Alfred Prufrock" and "Preludes," but chances are, I'm probably going to write more on Prufrock because a) it's longer and therefore has more material to work with and b) we've discussed it more in class. We've also been told that the 3-4 questions can be on lit techniques and devices, context, genre and generic conventions, language, discourse and ideology or reading strategies. I'm going to write some notes on some of these things, as normally the bit I find hardest is planning my essay. Once that's done, the writing normally takes care of itself.
Most of the stuff here is probably from class discussions, including from other people's notes (they were put on the J drive to be shared between us...). Yeah, it's plagiarism, I know, but a) I gave them my notes on Marxism and b) I'm not passing any of this stuff off as my own, I'm just putting it all here as a means to organise my thoughts. But if you really feel that strongly, I can get rid of some of the stuff here.
Context: Information
The socio-historical context: "The Love Song of J. Alfred Prufrock" was written in the early-ish 1900s (according to Wikipedia, Eliot started writing it in 1910, but it was published in 1915). This is a time of rapid change and turmoil- I mean, WWI happened at this time! Suddenly the whole world was plunged into a war, the likes of which had never been seen before, since the advent of new, powerful weapons as well as the usage of old, outdated tactics (i.e. charging in a straight line, which would have been useful for knights on horseback but not for modern soldiers in the face of machine guns) meant that loads of people were dying in the trenches pretty quickly. Also, as well as modern weaponry, other new inventions were changing the world: communications were improving via phones and TV, and transport was also improving.
Other trends present at this time were urbanisation and secularisation, though both of these kind of continue from previous times. During the Industrial Revolution of around 1760-1820, as well as afterwards, heaps of people began moving to the cities to find work, leading to crowded and busy cities which were sometimes dirty and polluted as greener technology had not yet been invented. The busy life in these crowded cities led to people becoming more disconnected from nature, and maybe even from each other too, despite the many advances in communication technology. The increase in secularisation at this time is simply yet another stage of a gradual increase of secularisation since the Renaissance period, but this time it was largely fuelled by all of the death and devastation caused by the World War which led people to challenge their faith.
T.S. Eliot: Okay, I must admit that I know hardly anything about this guy apart from the fact that he didn't discriminate; he hated pretty much everybody. But I do have a handout here about Eliot. Let's see what it says.
It says here that T.S. Eliot was born in St Louis (a city in Missouri, USA) in 1888, but his parents originally came from New England, which, believe it or not, is also a city in the USA (sort of shows just how overly patriotic those Brits were when they colonised half the world). It sort of ties in to how my Lit teacher was saying that Eliot is American but he likes pretending that he's British.
Anyway, Eliot grew up, discovered that he liked poetry, attended Harvard and studied some high-brow works by people like Omar Khayyam, Charles Baudelaire and Jules Laforgue. He also spent some time wandering around "the poorer and seedier parts of Boston" (yes, that's what the handout says). Apparently Charles Baudelaire is sometimes considered the father of Modernism, and from him, Eliot learned that he could write poetry on rather un-poetic material like industrial cities. From Jules Laforgue, a follower of Baudelaire, he learned about the French symbolist movement in poetry, as well as other random stuff like interior monologues which can be dialogues between rival aspects of the self, ironic voice, ironic mockery of society and self, irregularly placed rhymes, abrupt changes of setting, and more.
Your own context: ...
Context: How Prufrock is shaped by context
Socio-historical context: Industrialisation and its effects are seen in the way that the city has been portrayed as dilapidated, dirty and polluted as well as being pretty damn unwelcoming through word meanings and imagery. The streets are "muttering retreats," the word "muttering" suggesting discomfort because it's as if though there is some secret that you're not being let in on, and nobody likes not being let in on a secret. I guess you can also say that the muttering and the idea that there's something that you're not being let in on also hints at the idea of disconnection between humans. Further adding to this is the "insidious intent," that word "insidious" hinting that there really is something suspicious and unpleasant going on. The nights are also "restless" and uncomfortable, and the portrayal of the city with its "one-night cheap hotels" and "sawdust restaurants" (restaurants where the ground is covered with sawdust and all your rubbish is just dumped on the floor to be swept up with the sawdust at the end of the day) makes the city seem dirty and uncared for. And, of course, there's the imagery "yellow fog" and "yellow smoke," which hints at pollution and disease. Yellow is normally a happy colour, but when related to fog and smoke, it sounds almost diseased. Likewise, the smoke "rubs its back" and "rubs its muzzle" (kinaesthetic imagery! Yay, big words) like a cat or some other domestic creature, but unlike cats and dogs, when these descriptions are used to describe smoke, it makes the smoke seem almost too close for comfort. Oh, and this is kind of going back to what I said before about disconnection but there's also some other line about "lonely men in shirt sleeves," and that very word "lonely" creates a sense of disconnection, because isn't loneliness pretty much a lack of connection with other humans?
Fragmentation is another common theme of this time period because everything, including people's ideas, are all challenged to the limits with all the crazy stuff that's happening, like the World War, and, sure enough, there's a hell of a lot of fragmentation in this poem. First of all, there's all the crazy conflicting imagery, like "when the evening is spread out against the sky like a patient etherised upon a table." (Apparently there's a fancy word for abrupt transitions in style- "bathos." Unfortunately, I don't really know how to use it in a sentence.) I mean, look at that line (or rather, two lines). Just look at it. "When the evening is spread out upon the sky" sounds all Romantic-esque with its emphasis on nature and evening (which is when 90% of romantic scenes take place, I swear), but then it abruptly turns to that poor patient. The word "patient" suggests sickness and pain- a big difference from the romance of the evening and the sky. There's also sudden transitions in settings- one line Prufrock is "at dusk through narrow streets" and three lines later he is talking about how he "should have been a pair of ragged claws scuttling across the floors of silent seas." Even the word "ragged" sounds fragmented and disjointed in itself since it does mean "uneven" or "jagged." The structure of the poem is also pretty fragmented with its breaks (I'm not sure how to describe them- the dots or whatever that divides up sections of the poem).
Hm what else? I came up with some other idea when I was in the shower, but I've forgotten it now. Damn. Guess I'll have to revisit this part.
(10 minutes later...) I've remembered now. One thing that I didn't put in the information section above is that there were still pretty big divisions between the upper- and lower-classes, even during the World War. I remember reading some Horrible Histories book that said something about how the upper classes were complaining that they had to downgrade to having parlourmaids, while soldiers were being killed by the thousands in the trenches! Such class divisions are also pretty apparent in the poem- while the city, home of the working classes, is depicted as being dirty, unpleasant and possibly unsafe, the upper class life is much different. They have "tea and cakes and ices" while the lower classes have to make do with their "sawdust restaurants," and they have "sunsets and dooryards and sprinkled streets" while the lower classes have "streets that follow like a tedious argument." The juxtaposition between the two classes is also amplified by the constant switching between lower-class scenery and upper-class scenery.
T.S. Eliot's context: Okay, first of all, everything that my handout said that Eliot learned from reading Jules Laforgue can be found in this poem. "Interior monologues which are really ironic dialogues between rival aspects of the self" are found EVERYWHERE. In fact, you could almost argue that the entire poem is an "interior monologue which [is] really [an] ironic dialogue between rival aspects of the self." We can see this in all of his incessant questioning: "Do I dare?" "Do I dare disturb the universe?" "How should I presume?" "Should I then presume?" Most of these questions are probably directed towards himself, because normally "Do I dare?" is a question that you ask yourself, not other people. Also, the use of that stanza of Dante's Inferno at the beginning of the poem, which basically means "I can tell you my story only because I know you'll never return" suggests that this poem and its questions are not to be heard by anyone, and if they are to be heard, then only to a select audience chosen with care. There's also the bit about "time yet for a hundred indecisions, and for a hundred visions and revisions" which also creates an air of uncertainty. The very word "indecision" speaks for itself- there's not enough certainty to make a decision, and any decisions then have to face "revisions." Further examples of Prufrock's divided self appear in lines 39-45, where Prufrock initially makes a decision to "turn back and descend the stair" but then worries that other people will comment on "how his hair is growing thin," and then describes his attire as "rich and modest" before worrying that other people will comment on "how his arms and legs are thin." Here Prufrock is divided by his own judgement: one aspect of himself believing that he is prepared and ready to go, and the other rival aspect of himself only able to see his faults.
I don't really have too much to say on the next three dot points here- "air of worldly fatigue," "wry, ironic voice" and "ironic mockery of society and self." Now, the reason I don't have much to say is because there is nothing to say, because I know that there's lots to say, but because I don't know how to say it. I sort of get the "worldly fatigue" and "wry, ironic voice" in this poem, but not quite enough to provide solid examples. I guess the "wry, ironic voice" and "ironic mockery" might be in all that conflicting imagery with the evening being a "patient etherised on a table," as well as in some of the other random lines like "I have measured out my life with coffee spoons" (I dunno, that just felt "wry and ironic" to me), but I'm not sure how to explain how any of that stuff contributes to a "wry, ironic voice." In fact, I'm not sure if I really know the word "ironic" all too well. I mean, I know what the word means, but I don't know what it means... okay, maybe I should stop, I'm probably just confusing you. As for the "worldly fatigue," I guess that comes in when Prufrock asks if stuff "would be worth it" and talking about himself growing old. Maybe. I'm not sure.
The next two dot points here are "vers libre: reflecting the free flow of human consciousness as it attempts to come to terms with a complex reality" and "occasional, irregularly placed rhymes." Well, I can give examples for "occasional, irregularly placed rhymes." The first two lines rhyme, then the third line has no rhyme, then the next two lines rhyme, then "hotels" and "shells" sort of rhymes, then the next two lines rhyme, then there's a line which doesn't rhyme with any other line, then another two rhyming lines. Then there's that two line couplet thingy that rhymes. Likewise, there's some rhyming lines in the next stanza, but there's also some lines which inexplicably do not rhyme, like "licked its tongue into the corners of the evening," "and seeing that it was a soft October night," and so on. As for the vers libre thing, well, I'm not too sure on what the exact literary definition of "vers libre" is but I can see that this poem does appear to be a bit more "free" (I'm pretty sure that's what "libre" means) from most of the traditional rules of poetry and it is a reflection of the human consciousness. It leaps from topic to topic, a bit like how we can move from one train of thought to another (or maybe that's just me). It compares stuff to other random stuff ("patient etherised upon a table" anyone? Oh, wait, I've used that example about 3 times already), which I do too, and although I could be unique in that, I doubt it. It also has irregular line lengths, no fixed meter, irregular rhymes and random breaks (those dividing thingos- as I've said before I don't know what to call them...), because the human consciousness doesn't have a fixed meter, regular rhyme patterns and regular line lengths, not as far as I know anyway.
Next dot point- "abrupt changes of setting, timeframe and voice." Hm. I wonder where we can find those things in Prufrock? Here's a hint: everywhere. Abrupt changes of setting? "Streets that follow like a tedious argument" vs. "In the room the women come and go, talking of Michelangelo." "Gone at dusk through narrow streets" vs. "Scuttling across the floors of silent seas." Abrupt changes of timeframe? Not so obvious, but there is that bit that Prufrock randomly starts saying "I grow old... I grow old," and I doubt that he was old before because in the next line he says "I shall wear the bottom of my trousers rolled," the word "shall" indicating that he hasn't grown old yet. Abrupt changes in voice? Should I pull out my favourite quote again? Nah, you're probably all sick of it by now.
Now for the dot point that sprouts several dot points: "Laforgian symbolism." First up is "A kind of poetry which would be psychology in the form of a dream... with flowers and scents and wind... complex symphonies with certain phrases (motifs) returning from time to time." Now, that language is too flowery and poetic for my puny brain, but I do understand the bit about certain phrases returning from time to time. "The Love Song of J. Alfred Prufrock" has some of these, such as "In the room the women come and go/ Talking of Michelangelo," "Do I dare? and "And would it have been worth it, after all." Now, we could dissect these further by talking about the effect that this repetition has on our readings of the poem and whatnot, but I can't be bothered doing that now. The next dot-point-within-a-dot-point is "Images which are absolutely precise in physical terms but endlessly suggestive in their meanings," which I think is just like that patient... You probably know that quote by now, even if you've never read the poem before. There's also the "pinned and wriggling on the wall" thing, which is pretty precise, but it can also mean other stuff like being trapped. Trapped by what? That's up to you to decide. "Building up of patterns of meaning through juxtaposition and accumulation of images," the third dot point, is seen through the juxtaposition between the upper- and lower- classes' descriptions, as I've said before. Finally, there's "contrast of sublime and banal images." I originally thought that "banal" meant something along the lines of "gruesome" or "unpleasant," like that quote that I've repeated oh-so-many times, but no, apparently it means "so lacking in originality as to be obvious and boring." I can't really say that much of the imagery in this poem is "lacking in originality," but I guess there is the "magic lantern [throwing] the nerves in patterns on a screen" followed by the more mundane "settling a pillow or throwing off a shawl."
That's all the dot points for that one covered now. Phew!
Now, I was going to cover multiple things in this post, but this post has been going on for quite a while now so I'm going to talk about this poem in multiple blog posts. Stay tuned...
The Baroque Period (c.1600-1750) - Bach's Brandenburg Concerto no. 2 (c. 1717-1723)
It's been a while since I've last written about music, but now I'd like to expand on what I talked about last year, especially with regards to the Baroque period. Last year, I wrote about what society was like during the Baroque era and Handel's Messiah, a very famous oratorio from the Baroque period. This time, I'm going to write about another famous work by another famous Baroque composer: Bach's Brandenburg Concerto no. 2, one of a collection of six concertos written for the margrave (which Google says is "the hereditary title of some princes of the Holy Roman Empire") of Brandenburg, one of the states of Bach's home country of Germany.
The only movements that Music 3AB students have to study are the first two, but it doesn't hurt to be reasonably familiar with the third either. I'm only going to talk about the first two movements for now though.
Bach's Brandenburg Concerto no. 2 is, of course, of the concerto genre. Basically, concertos draw on the whole idea of the stile concertato, or rather, the principle of strong contrasts within a piece of music. Concertos normally achieve this contrast by pitting a soloist or a group of soloists against the rest of the ensemble (known as the ripieno). Concertos in which the group is divided up into a soloist and ripieno is called a solo concerto, while concertos in which the group is divided up into a small group of soloists (a.k.a. the "concertino") and the ripieno is known as a concerto grosso. Bach's Brandenburg Concerto no. 2 is an example of a concerto grosso, with a concertino consisting of a high F trumpet, a flute, an oboe and a violin, as well as a ripieno containing strings and basso continuo. (Basso continuo is basically the group of instruments playing the bass line of Baroque music, and normally consists of one bass instrument and a keyboard instrument; for example, a basso continuo can consist of a cello and a harpsichord.)
Just before I get into the nitty-gritty details of the piece, let's have a quick look at Bach's musical career as a whole. Bach, working as a musician in the Baroque period, needed to have employment to make a living. From the age of 18, he started working as an organist to the New Church in Arnstadt, and later at Mühlhausen. His employment at the time did not require him to compose anything, but he composed a few cantatas nevertheless. Later on, when he wanted to get into composing church music but his employers at Mühlhausen weren't too keen on church music, Bach looked for work elsewhere, and was fortunate enough to find a position as a member of the chamber orchestra and organist at the Ducal Court at Weimar. At this point, he was still much better known for his instrumental skills than his composing skills. In 1714, however, Bach was promoted to Concertmaster, where he was required to produce one cantata a month. In 1717, Bach was offered an even better position at the Court of Anhalt-Cöthen: the position of Capellmeister, the highest rank given to a musician during the Baroque period. Unfortunately, his employers here also weren't so keen on church music, so Bach composed more secular cantatas and instrumental works instead. He also composed the 6 Brandenburg Concertos at this time. In 1723, Bach finally got his dream job, supervising the music for Leipzig's four main churches. The work was very demanding as it also involved training the choristers, but he must have liked it because he stayed in this position for some 27 years- until the day he died.
Now, where were we? Oh, yes, that's right- back to the music. Let's start off with the first movement of Bach's Brandenburg Concerto no. 2- it seems a rather logical place to start.
This movement is in ritornello form, which is kind of like rondo form in which there is a recurring theme, or ritornello, that is interspersed with other themes. However, unlike rondo form in which the recurring theme is repeated almost exactly the same way every time, the recurring theme can change over time: it can change key or be extended and developed in other ways. Alternatively, it might also be fragmented: only part of the ritornello may appear.
Now for a really quick analysis of this piece.
The ritornello theme in this piece is heard for the first 8 bars, after which there is an episode in which the solo violin plays a tune (accompanied by basso continuo, of course- basso continuo ties the work together). Then everyone comes back in for the ritornello for another couple of bars, before the solo oboe plays the same tune, accompanied by solo violin and basso continuo. Then pretty much the same thing happens twice more except that we've now modulated to C Major and the combination of instruments in the episodes changes: in the next episode it's solo flute melody with solo oboe and basso continuo accompaniment, and in the episode after that it's solo trumpet with solo flute and basso continuo accompaniment. Next everyone comes back in for the ritornello, except this time the ritornello occurs for 6 bars, as opposed to all those puny little 2-bar fragments that we had before. Then the entire concertino plays in the next episode, though only the trumpet has the melody: all the other soloists and the basso continuo are just accompaniment. Everyone then comes back for the ritornello, but now we're in D minor for a ritornello that goes on for a while and has a whole bunch of sequences and some sequential imitation in it. A sequence, by the way, is basically a small melodic fragment that repeats over and over again at different pitches. Imitation is when a melodic fragment is repeated by different instruments. Anyway, as the ritornello goes on, after a few sequences, we come to a part where the melody is passed between different instruments, kind of like imitation as I said above. First it's played by solo trumpet, then solo flute, then solo oboe, and then the music modulates back to the home key for a while before finally we get to another episode, this time a solo flute playing the melody, accompanied by solo violin and basso continuo. Soon solo oboe joins in with the melody while flute gets delegated to an accompaniment line (I don't know if it's just me, but I think that it sort of hints of the movement following this one). Finally, the trumpet joins in with the melody while oboe becomes yet another accompaniment line, followed by a new ritornello in what I think is C minor but I'm not sure (I can't find the sheet where I wrote down all the modulations and stuff, but I'm not exactly looking hard right now, so I might find it later). This is yet another pretty long ritornello with sequences and terraced dynamics and stuff as well as something called "stretto" in bar 94 which is where the time between entries in a canon or whatever is reduced (e.g. instead of 4 beats between entries, it might be only 2 beats instead). At last, in the upbeat to bar 103 onwards, we hear the ritornello tune for the last time. First, we hear it unison, then everyone goes off and does their own thing after only a few bars (typical Baroque music...). The piece comes to an end in bar 118, without a rit or anything, probably because that would spoil the drive and forward motion so prized during the Baroque era.
Wow. That was a lot. *exhales deeply and noisily*
Now for the second movement!
The second movement only has the concertino instruments minus the trumpet, and the basso continuo. The ripieno strings aren't playing here. This movement is in D minor, as opposed to F Major of the movement before, which is partly why the trumpet isn't playing in this movement- in the Baroque era, brass instruments could only really play well in one particular key because they didn't have all the valves and whatnot that modern brass instruments have today. In fact, many middle movements of concertos lack trumpets because the middle movement tends to be in a different key, as well as being generally softer and slower (so you wouldn't want some loud bright trumpet spoiling the whole effect).
Anyway, pretty much the whole movement is a round. The violin starts off with the melody, and then the oboe joins in two bars later, followed by the flute that joins in two bars later. It's not an exact round, as the flute part in bars 19-23 doesn't exactly mirror anything that has happened in the oboe or violin parts. The basso continuo is pretty much just playing broken chords, which is great if you want to be a nerd and practise doing some harmonic analysis.
I don't really have much more in the way of analysis for this movement, but do listen to it because it's pretty.
Now, after hearing this, you might be curious about the other Brandenburgs. All the other Brandenburg Concertos had different sets of instruments making up the concertino. Some of these Brandenburg Concertos are also classified as "orchestral concertos" or something like that in which there isn't really a defined split between concertino and ripieno.
No. 1: Violino piccolo, 3 oboes, 2 horns, bassoon
No. 2: Violin, flute, oboe, trumpet
No. 3: Strings
No. 4: Violin, 2 flutes
No. 5: Clavier (harpsichord), violin, flute
No. 6: Violone, 2 violas da gamba, viols with 6 strings (these are all low string instruments from the Baroque era)
No. 5 has a really crazy harpsichord solo. There is a pretty damn strong contrast between the style of the harpsichord solo and the rest of the work. I guess it fits with the whole stile concertato and principle of contrast?
The only movements that Music 3AB students have to study are the first two, but it doesn't hurt to be reasonably familiar with the third either. I'm only going to talk about the first two movements for now though.
Bach's Brandenburg Concerto no. 2 is, of course, of the concerto genre. Basically, concertos draw on the whole idea of the stile concertato, or rather, the principle of strong contrasts within a piece of music. Concertos normally achieve this contrast by pitting a soloist or a group of soloists against the rest of the ensemble (known as the ripieno). Concertos in which the group is divided up into a soloist and ripieno is called a solo concerto, while concertos in which the group is divided up into a small group of soloists (a.k.a. the "concertino") and the ripieno is known as a concerto grosso. Bach's Brandenburg Concerto no. 2 is an example of a concerto grosso, with a concertino consisting of a high F trumpet, a flute, an oboe and a violin, as well as a ripieno containing strings and basso continuo. (Basso continuo is basically the group of instruments playing the bass line of Baroque music, and normally consists of one bass instrument and a keyboard instrument; for example, a basso continuo can consist of a cello and a harpsichord.)
Just before I get into the nitty-gritty details of the piece, let's have a quick look at Bach's musical career as a whole. Bach, working as a musician in the Baroque period, needed to have employment to make a living. From the age of 18, he started working as an organist to the New Church in Arnstadt, and later at Mühlhausen. His employment at the time did not require him to compose anything, but he composed a few cantatas nevertheless. Later on, when he wanted to get into composing church music but his employers at Mühlhausen weren't too keen on church music, Bach looked for work elsewhere, and was fortunate enough to find a position as a member of the chamber orchestra and organist at the Ducal Court at Weimar. At this point, he was still much better known for his instrumental skills than his composing skills. In 1714, however, Bach was promoted to Concertmaster, where he was required to produce one cantata a month. In 1717, Bach was offered an even better position at the Court of Anhalt-Cöthen: the position of Capellmeister, the highest rank given to a musician during the Baroque period. Unfortunately, his employers here also weren't so keen on church music, so Bach composed more secular cantatas and instrumental works instead. He also composed the 6 Brandenburg Concertos at this time. In 1723, Bach finally got his dream job, supervising the music for Leipzig's four main churches. The work was very demanding as it also involved training the choristers, but he must have liked it because he stayed in this position for some 27 years- until the day he died.
Now, where were we? Oh, yes, that's right- back to the music. Let's start off with the first movement of Bach's Brandenburg Concerto no. 2- it seems a rather logical place to start.
This movement is in ritornello form, which is kind of like rondo form in which there is a recurring theme, or ritornello, that is interspersed with other themes. However, unlike rondo form in which the recurring theme is repeated almost exactly the same way every time, the recurring theme can change over time: it can change key or be extended and developed in other ways. Alternatively, it might also be fragmented: only part of the ritornello may appear.
Now for a really quick analysis of this piece.
The ritornello theme in this piece is heard for the first 8 bars, after which there is an episode in which the solo violin plays a tune (accompanied by basso continuo, of course- basso continuo ties the work together). Then everyone comes back in for the ritornello for another couple of bars, before the solo oboe plays the same tune, accompanied by solo violin and basso continuo. Then pretty much the same thing happens twice more except that we've now modulated to C Major and the combination of instruments in the episodes changes: in the next episode it's solo flute melody with solo oboe and basso continuo accompaniment, and in the episode after that it's solo trumpet with solo flute and basso continuo accompaniment. Next everyone comes back in for the ritornello, except this time the ritornello occurs for 6 bars, as opposed to all those puny little 2-bar fragments that we had before. Then the entire concertino plays in the next episode, though only the trumpet has the melody: all the other soloists and the basso continuo are just accompaniment. Everyone then comes back for the ritornello, but now we're in D minor for a ritornello that goes on for a while and has a whole bunch of sequences and some sequential imitation in it. A sequence, by the way, is basically a small melodic fragment that repeats over and over again at different pitches. Imitation is when a melodic fragment is repeated by different instruments. Anyway, as the ritornello goes on, after a few sequences, we come to a part where the melody is passed between different instruments, kind of like imitation as I said above. First it's played by solo trumpet, then solo flute, then solo oboe, and then the music modulates back to the home key for a while before finally we get to another episode, this time a solo flute playing the melody, accompanied by solo violin and basso continuo. Soon solo oboe joins in with the melody while flute gets delegated to an accompaniment line (I don't know if it's just me, but I think that it sort of hints of the movement following this one). Finally, the trumpet joins in with the melody while oboe becomes yet another accompaniment line, followed by a new ritornello in what I think is C minor but I'm not sure (I can't find the sheet where I wrote down all the modulations and stuff, but I'm not exactly looking hard right now, so I might find it later). This is yet another pretty long ritornello with sequences and terraced dynamics and stuff as well as something called "stretto" in bar 94 which is where the time between entries in a canon or whatever is reduced (e.g. instead of 4 beats between entries, it might be only 2 beats instead). At last, in the upbeat to bar 103 onwards, we hear the ritornello tune for the last time. First, we hear it unison, then everyone goes off and does their own thing after only a few bars (typical Baroque music...). The piece comes to an end in bar 118, without a rit or anything, probably because that would spoil the drive and forward motion so prized during the Baroque era.
Wow. That was a lot. *exhales deeply and noisily*
Now for the second movement!
The second movement only has the concertino instruments minus the trumpet, and the basso continuo. The ripieno strings aren't playing here. This movement is in D minor, as opposed to F Major of the movement before, which is partly why the trumpet isn't playing in this movement- in the Baroque era, brass instruments could only really play well in one particular key because they didn't have all the valves and whatnot that modern brass instruments have today. In fact, many middle movements of concertos lack trumpets because the middle movement tends to be in a different key, as well as being generally softer and slower (so you wouldn't want some loud bright trumpet spoiling the whole effect).
Anyway, pretty much the whole movement is a round. The violin starts off with the melody, and then the oboe joins in two bars later, followed by the flute that joins in two bars later. It's not an exact round, as the flute part in bars 19-23 doesn't exactly mirror anything that has happened in the oboe or violin parts. The basso continuo is pretty much just playing broken chords, which is great if you want to be a nerd and practise doing some harmonic analysis.
I don't really have much more in the way of analysis for this movement, but do listen to it because it's pretty.
Now, after hearing this, you might be curious about the other Brandenburgs. All the other Brandenburg Concertos had different sets of instruments making up the concertino. Some of these Brandenburg Concertos are also classified as "orchestral concertos" or something like that in which there isn't really a defined split between concertino and ripieno.
No. 1: Violino piccolo, 3 oboes, 2 horns, bassoon
No. 2: Violin, flute, oboe, trumpet
No. 3: Strings
No. 4: Violin, 2 flutes
No. 5: Clavier (harpsichord), violin, flute
No. 6: Violone, 2 violas da gamba, viols with 6 strings (these are all low string instruments from the Baroque era)
No. 5 has a really crazy harpsichord solo. There is a pretty damn strong contrast between the style of the harpsichord solo and the rest of the work. I guess it fits with the whole stile concertato and principle of contrast?
Friday, April 5, 2013
Complex Numbers and Polar Coordinates
(This post was cut and pasted from one of my other blogs.)
So... I have this Maths Specialist investigation on complex numbers in polar form, and they say that the best way to learn something is to teach other people, which is what I'm going to do!
These posts really need diagrams... but I'm pretty lazy, so I'll just add diagrams when I feel like it/ when you pressure me to :)
ANYWAY, I'll get to the point.
Polar Coordinates
Most of the time, when you are given coordinates for a graph, you are given the coordinates in rectangular form. Rectangular form is basically (x, y) where x is the number of units to the right and y is the number of units up. Polar coordinates differ quite drastically: they give coordinates in the form (r,θ) where r is the distance from the origin, and θ is the angle anticlockwise from the positive x-axis.
So, how do you go about converting rectangular coordinates to polar form? It's simple...
To find r, the direct distance from the origin, all you have to do is use Pythagoras' theorem. According to Pythagoras' theorem, r^2 = x^2 + y^2, so r is the square root of x^2 + y^2. (This would all make a lot more sense if I put a graph here, but as I said, I'm pretty lazy.)
Finding the angle between r and the x-axis takes a bit more work. Using SOHCAHTOA (sin = opp/hyp, cos = adj/hyp, tan = opp/adj), tan θ is equal to y/x. (Once again, it would be a lot easier to see if I could be bothered drawing a graph.) Therefore, θ is simply the reverse tangent of y/x.
The above was for if the point (x,y) is in the first quadrant of the graph (i.e. x and y are positive). But what happens if the point (x,y) is in a different section of the graph? No worries- you may simply have to add or subtract 180 or 360 to give the anticlockwise angle from the point (x,y) and the positive x-axis. If the graph lies in the 2nd quadrant (i.e. x is negative, y is positive), you need to subtract the angle from 180 degrees. If the graph lies in the 3rd quadrant (i.e. both x and y are negative) then you need to add 180 degrees. If the graph lies in the 4th quadrant (i.e. x is positive, y is negative), then you need to subtract the angle from 360 degrees. If you ever get confused, draw a graph and draw an angle anticlockwise starting from the positive x-axis. Diagrams are a very useful tool for helping your understanding.
By the way, there are two acceptable formats for giving an angle: the angle anticlockwise from the positive x-axis is given as a positive angle, while the angle clockwise from the positive x-axis is given as a negative angle. Therefore, the angle 270 degrees can also be written as -90 degrees.
To convert from polar form to standard Cartesian coordinates, it helps to draw a quick sketch of the triangle with its hypotenuse and angle marked. You can then easily use SOHCAHTOA to find x and y.
Complex Numbers
Complex numbers are made up of two parts: a real part and an imaginary part. They take on the form z = x + yj (i is sometimes substituted for j, but they both mean the same thing). x is the real part, and yj is the imaginary part. j (or i) is equal to the square root of negative one (which is a real number), but x and y themselves are real numbers- one example of a complex number is 5 + 3j.
Let's take any real number and call it a. If you have the square root of a, and you raise it to the power of 2, you end up with a. Similarly, if you take the square root of -1, and you raise it to the power of 2, you should theoretically also end up with -1. As j = -1, j^2 must also equal -1. This is important for complex number arithmetic.
Adding, subtracting and multiplying complex numbers is easy. When adding or subtracting, just add or subtract the real parts together and then add or subtract the imaginary parts. When multiplying, just multiply as you would normally do in algebra. However, if you end up with j^2, make sure to substitute -1 in its place. (Additional substitution may be required for higher powers of j.)
Dividing requires a new concept: the conjugate of a complex number.
Basically, to get the conjugate of a complex number, just change the addition sign to a subtraction sign and vice versa.
When dividing complex numbers, multiply by the conjugate of the bottom number over itself. For example, if you want to do (2 + 3j)/(4 - 6j), multiply by (4 + 6j)/(4 + 6j). This works for two reasons:
Reason 1: Difference of Two Squares.
When you multiply the bottom number by its conjugate, you get a difference of two squares expression. (x + yj)(x - yj) = x^2 + xyj - xyj - (yj)^2. xyj and -xyj then cancel each other out and the resulting expression can be expanded to x^2-(y^2)(j^2). As I have said earlier, j^2 = -1, so (x + yj)(x - yj) = x^2 + y^2 which is a real number! Therefore, after multiplying a fraction by the conjugate of the bottom number over itself, you really end up simply dividing by a real number at the end.
Reason 2: The Conjugate Over Itself is Equal to 1.
When you multiply the fraction by the conjugate of the bottom number over itself, you simplify it so that you are dividing by a real number, but you are not changing the value of the expression. This is because any number over itself is equal to 1. Therefore, when you are multiplying by the conjugate over itself, the value of the expression remains unchanged.
Complex numbers are able to be graphed. The name for the graph is called the Argand diagram but it's actually very similar to any other graph. The main difference is that the x-axis and y-axis are given different names: the x-axis is the real axis and the y-axis is the imaginary axis. To find the location complex number x + yj on the Argand diagram, simply move x units to the right and y units up.
Now on to the next part...
Complex Numbers in Polar Form
Unfortunately, converting complex numbers to polar form isn't as easy as converting normal numbers to polar form. I think that this is because a complex number has two components (real and imaginary), while a normal number only has one real component. Nevertheless, it's not that hard.
First, we have to use a little trigonometry (again, I need a diagram here). Let r be the distance from the origin to the point x + yj on the Argand diagram and θ be the angle counterclockwise between the positive real axis and r.
cos θ = adj/hyp = x / r. Therefore x = r cos θ.
sin θ = opp/hyp = y / r. Therefore y = r sin θ.
Substituting these values into the general complex number form z = x + yj you get:
z = r cos θ + (r sin θ)j = r (cos θ + j sin θ)
Now, using Pythagoras, r = sqrt(x^2 + y^2). Now |z| = sqrt(z multiplied by the conjugate of z). As I stated in the above section about complex numbers, z multiplied by its conjugate is x^2 + y^2. Therefore, |z| = r = sqrt(x^2 + y^2).
Therefore, z = |z| (cos θ + j sin θ).
Some terminology for you: |z| is the modulus of the complex number. θ is the argument (or arg).
Now let's move on to... the powers of complex numbers in polar form.
No, I didn't mean that they had powers, I meant raising them to a power. Don't get too excited.
When you raise z to a power p, the modulus also gets raised to a power p. The argument, on the other hand, simply gets multiplied by the power. It's like the log laws!
log (z^p) = p log z
arg (z^p) = p arg z
Multiplying and Dividing Complex Numbers
Multiplying and dividing complex numbers in polar form are like raising complex numbers in polar form to powers. The modulus gets treated like a normal number while you have to treat the argument like a logarithm.
When multiplying two complex numbers, say z and w, the modulus is simply |z|*|w| = |zw|. The argument is simply arg (z) + arg (w) = arg (zw).
When dividing two complex numbers, say z and w, the modulus is simply |z|/|w| = |z/w|. The argument is simply arg (z) - arg (w) = arg (z/w).
I think that the laws about multiplying, dividing and raising the modulus to a power are the same as those around normal numbers because the modulus is an ordinary real number as well. But why are the above laws surrounding arguments the same as the laws surrounding logs? Let's take a brief glance at two trigonometry laws and see how we can use them to explain multiplying (and possibly dividing too). These laws are:
cos (θ + φ) = cos θ cos φ - sin θ sin φ
sin (θ + φ) = sin θ cos φ + cos θ sin φ
Let z = |z|(cos θ + j sin θ)
Let w = |w|(cos φ + j sin φ)
zw = |z|(cos θ + j sin θ)|w|(cos φ + j sin φ)
= |zw|(cos θcos φ + i sin θcos φ + cos θ i sin φ + i sin θ i sin φ)
= |zw|((cos θcos φ - sin θsin φ)+ i(sin θcos φ + cos θsin φ))
Using the laws stated previously,
zw = |zw|(cos (θ + φ) + sin (θ + φ))
The modulus of z and w have been multiplied together and the arguments have been added together.
There's a website on how to explain why arg(z^p) = p arg (z), but it looks too confusing for me. If you're interested, though, you can find it at http://math.tutorvista.com/trigonometry/de-moivre-s-theorem.html .
So... I have this Maths Specialist investigation on complex numbers in polar form, and they say that the best way to learn something is to teach other people, which is what I'm going to do!
These posts really need diagrams... but I'm pretty lazy, so I'll just add diagrams when I feel like it/ when you pressure me to :)
ANYWAY, I'll get to the point.
Polar Coordinates
Most of the time, when you are given coordinates for a graph, you are given the coordinates in rectangular form. Rectangular form is basically (x, y) where x is the number of units to the right and y is the number of units up. Polar coordinates differ quite drastically: they give coordinates in the form (r,θ) where r is the distance from the origin, and θ is the angle anticlockwise from the positive x-axis.
So, how do you go about converting rectangular coordinates to polar form? It's simple...
To find r, the direct distance from the origin, all you have to do is use Pythagoras' theorem. According to Pythagoras' theorem, r^2 = x^2 + y^2, so r is the square root of x^2 + y^2. (This would all make a lot more sense if I put a graph here, but as I said, I'm pretty lazy.)
Finding the angle between r and the x-axis takes a bit more work. Using SOHCAHTOA (sin = opp/hyp, cos = adj/hyp, tan = opp/adj), tan θ is equal to y/x. (Once again, it would be a lot easier to see if I could be bothered drawing a graph.) Therefore, θ is simply the reverse tangent of y/x.
The above was for if the point (x,y) is in the first quadrant of the graph (i.e. x and y are positive). But what happens if the point (x,y) is in a different section of the graph? No worries- you may simply have to add or subtract 180 or 360 to give the anticlockwise angle from the point (x,y) and the positive x-axis. If the graph lies in the 2nd quadrant (i.e. x is negative, y is positive), you need to subtract the angle from 180 degrees. If the graph lies in the 3rd quadrant (i.e. both x and y are negative) then you need to add 180 degrees. If the graph lies in the 4th quadrant (i.e. x is positive, y is negative), then you need to subtract the angle from 360 degrees. If you ever get confused, draw a graph and draw an angle anticlockwise starting from the positive x-axis. Diagrams are a very useful tool for helping your understanding.
By the way, there are two acceptable formats for giving an angle: the angle anticlockwise from the positive x-axis is given as a positive angle, while the angle clockwise from the positive x-axis is given as a negative angle. Therefore, the angle 270 degrees can also be written as -90 degrees.
To convert from polar form to standard Cartesian coordinates, it helps to draw a quick sketch of the triangle with its hypotenuse and angle marked. You can then easily use SOHCAHTOA to find x and y.
Complex Numbers
Complex numbers are made up of two parts: a real part and an imaginary part. They take on the form z = x + yj (i is sometimes substituted for j, but they both mean the same thing). x is the real part, and yj is the imaginary part. j (or i) is equal to the square root of negative one (which is a real number), but x and y themselves are real numbers- one example of a complex number is 5 + 3j.
Let's take any real number and call it a. If you have the square root of a, and you raise it to the power of 2, you end up with a. Similarly, if you take the square root of -1, and you raise it to the power of 2, you should theoretically also end up with -1. As j = -1, j^2 must also equal -1. This is important for complex number arithmetic.
Adding, subtracting and multiplying complex numbers is easy. When adding or subtracting, just add or subtract the real parts together and then add or subtract the imaginary parts. When multiplying, just multiply as you would normally do in algebra. However, if you end up with j^2, make sure to substitute -1 in its place. (Additional substitution may be required for higher powers of j.)
Dividing requires a new concept: the conjugate of a complex number.
Basically, to get the conjugate of a complex number, just change the addition sign to a subtraction sign and vice versa.
When dividing complex numbers, multiply by the conjugate of the bottom number over itself. For example, if you want to do (2 + 3j)/(4 - 6j), multiply by (4 + 6j)/(4 + 6j). This works for two reasons:
Reason 1: Difference of Two Squares.
When you multiply the bottom number by its conjugate, you get a difference of two squares expression. (x + yj)(x - yj) = x^2 + xyj - xyj - (yj)^2. xyj and -xyj then cancel each other out and the resulting expression can be expanded to x^2-(y^2)(j^2). As I have said earlier, j^2 = -1, so (x + yj)(x - yj) = x^2 + y^2 which is a real number! Therefore, after multiplying a fraction by the conjugate of the bottom number over itself, you really end up simply dividing by a real number at the end.
Reason 2: The Conjugate Over Itself is Equal to 1.
When you multiply the fraction by the conjugate of the bottom number over itself, you simplify it so that you are dividing by a real number, but you are not changing the value of the expression. This is because any number over itself is equal to 1. Therefore, when you are multiplying by the conjugate over itself, the value of the expression remains unchanged.
Complex numbers are able to be graphed. The name for the graph is called the Argand diagram but it's actually very similar to any other graph. The main difference is that the x-axis and y-axis are given different names: the x-axis is the real axis and the y-axis is the imaginary axis. To find the location complex number x + yj on the Argand diagram, simply move x units to the right and y units up.
Now on to the next part...
Complex Numbers in Polar Form
Unfortunately, converting complex numbers to polar form isn't as easy as converting normal numbers to polar form. I think that this is because a complex number has two components (real and imaginary), while a normal number only has one real component. Nevertheless, it's not that hard.
First, we have to use a little trigonometry (again, I need a diagram here). Let r be the distance from the origin to the point x + yj on the Argand diagram and θ be the angle counterclockwise between the positive real axis and r.
cos θ = adj/hyp = x / r. Therefore x = r cos θ.
sin θ = opp/hyp = y / r. Therefore y = r sin θ.
Substituting these values into the general complex number form z = x + yj you get:
z = r cos θ + (r sin θ)j = r (cos θ + j sin θ)
Now, using Pythagoras, r = sqrt(x^2 + y^2). Now |z| = sqrt(z multiplied by the conjugate of z). As I stated in the above section about complex numbers, z multiplied by its conjugate is x^2 + y^2. Therefore, |z| = r = sqrt(x^2 + y^2).
Therefore, z = |z| (cos θ + j sin θ).
Some terminology for you: |z| is the modulus of the complex number. θ is the argument (or arg).
Now let's move on to... the powers of complex numbers in polar form.
No, I didn't mean that they had powers, I meant raising them to a power. Don't get too excited.
When you raise z to a power p, the modulus also gets raised to a power p. The argument, on the other hand, simply gets multiplied by the power. It's like the log laws!
log (z^p) = p log z
arg (z^p) = p arg z
Multiplying and Dividing Complex Numbers
Multiplying and dividing complex numbers in polar form are like raising complex numbers in polar form to powers. The modulus gets treated like a normal number while you have to treat the argument like a logarithm.
When multiplying two complex numbers, say z and w, the modulus is simply |z|*|w| = |zw|. The argument is simply arg (z) + arg (w) = arg (zw).
When dividing two complex numbers, say z and w, the modulus is simply |z|/|w| = |z/w|. The argument is simply arg (z) - arg (w) = arg (z/w).
I think that the laws about multiplying, dividing and raising the modulus to a power are the same as those around normal numbers because the modulus is an ordinary real number as well. But why are the above laws surrounding arguments the same as the laws surrounding logs? Let's take a brief glance at two trigonometry laws and see how we can use them to explain multiplying (and possibly dividing too). These laws are:
cos (θ + φ) = cos θ cos φ - sin θ sin φ
sin (θ + φ) = sin θ cos φ + cos θ sin φ
Let z = |z|(cos θ + j sin θ)
Let w = |w|(cos φ + j sin φ)
zw = |z|(cos θ + j sin θ)|w|(cos φ + j sin φ)
= |zw|(cos θcos φ + i sin θcos φ + cos θ i sin φ + i sin θ i sin φ)
= |zw|((cos θcos φ - sin θsin φ)+ i(sin θcos φ + cos θsin φ))
Using the laws stated previously,
zw = |zw|(cos (θ + φ) + sin (θ + φ))
The modulus of z and w have been multiplied together and the arguments have been added together.
There's a website on how to explain why arg(z^p) = p arg (z), but it looks too confusing for me. If you're interested, though, you can find it at http://math.tutorvista.com/trigonometry/de-moivre-s-theorem.html .
Very brief review of chapters 1-6 (Sadler 3CD Spec) - Part 2
Well, since I didn't get anywhere near to finishing part 1 last time, here's part 2.
Before I get started, I was really curious to find out how people know about this blog apart from the people I've told, because I would have thought you'd have to be pretty specific in your searches to get this particular blog in amongst all of the other websites displaying the same sort of information. One thing that came up was this website here: http://library.onlinepatashala.com/SubPlay.aspx?Video=admin@2509. This blog is listed under the list of blogs in the bottom right corner. Seems a bit odd that a blog like this, written in such an obnoxious style, can be listed on an educational website like that. Maybe I should make a poll or something to find out how everyone's finding out about this blog because I am one of those annoying overly curious people.
Anyway, time to get started on speed reviewing chapters 3 to 6!
Basics of 3D Vectors
3D vectors are pretty similar to 2D vectors, apart from the fact that they're obviously in 3D and as such have an extra component for the extra dimension. Instead of good ol' ai + bj, now you're dealing with ai + bj + ck. Pretty much everything else is the same, though- adding, subtracting and finding the magnitude are all the same. Yes, even in 3D you're finding sqrt(a^2 + b^2 + c^2). The procedure for working out the unit vector is also the same, as is the procedure for finding the angle between two lines. (Getting bored of this now, are you?) Yes, I need to write a blog post on how to do all this stuff, but as I said before, this is a brief review so I'm assuming that you know stuff like that from last year's Spec work.
(By the way, check out Sadler's example at the beginning of the section, particularly the guy called Captain Over. "Hi Victor, it's Over, over?" Almost as bad as Rosencrantz and Guildenstern- "I think we're off course." "Of course!" I'm not sure if that's the exact wording but I can't be bothered checking.)
Equations of Lines and Planes
The vector equation of a line in 3D works exactly the same as the equation of a line in 2D. Basically, you have the position vector of one point in the line plus a scalar multiple of the direction of the line, in the form r = a + (lambda)b.
The normal equation of a line in 3D unfortunately doesn't work the same way, as every line has multiple possible lines that are perpendicular to it. The way I helped myself visualise this one was by sticking my arm out and then holding my ruler perpendicular to my arm, and then seeing for myself that there are multiple ways in which this could be achieved. Maybe I'm just weird, or maybe it might help you visualise it too.
However, you can have a normal equation of a plane. This is because a plane (a.k.a. a big flat thing that extends in all directions, NOT the aeroplanes that fly, unfortunately) only has one direction that's perpendicular to it. Therefore, you can use the equation r (dot) n = c where c = a (dot) n, where a is a point on the plane and n is a vector perpendicular to the plane. To find out a vector perpendicular to a plane, you can do several things.
1) Sometimes you might get lucky and they'll say that the plane is perpendicular to a certain vector/plane/whatever. Then you can just use the vector normal that they give you!
2) If you get a vector parallel to the plane in question, then you need to find a vector perpendicular to that vector (i.e. if it's perpendicular to a plane parallel to your plane, then it's also perpendicular to your plane. You can see this in my post about Scalar Product and Closest Approach). To do this, you can use dot product and guess and check. I normally let the vector normal be equal to xi + yj + zk and then set the dot product of that and the parallel vector equal to 0. Then I let one of the variables be equal to 0 and then work out what y is in relation to z (for example, y = z or 2x = z). Then I pick numbers that suit that criteria (e.g. for y = z I might choose to make x = 0, y = 1 and z = 1, and for 2x = z I might make it x = 1, y = 0 and z = 2) and double check to make sure that I've wound up with a vector normal to the plane.
3) If you get given two lines, you need to find a vector normal to both lines. Find the dot product between each line and xi + yj + zk, and you'll wind up with two equations for x, y and z. This isn't enough to give you exact numbers, but it is enough to give you the relationship between the variables if you tweak your equations around a bit and/or set one of the variables equal to 0.
4) If you get given three points, then you need to find two lines by finding the position of one point relative to the other two. Then, just use the methods mentioned above.
By the way, there are also vector and Cartesian equations of a plane. The vector equation is in the form r = a + (lambda)b + (mu)c, where b and c are non-parallel vectors that are parallel to the plane. The Cartesian form is simply in the form ax + by + cz = (constant). One great thing about this form is that ai + bj + ck is a vector perpendicular to whatever plane you're looking at.
Interception between lines and planes are worked out in the same way as 2D vectors.
Quick Notes on Proofs by Deduction
"Proof by Deduction" is basically the fancy term for most of the proofs that you've probably been doing so far- you know, the ones where you say that "this equals this, because of that axiom" or whatever and you keep putting down logical facts and explanations until you prove whatever it is that had to be proved? Okay, terrible explanation. Maybe I should provide an example: trig proofs. Proofs where you say "a equals e, because they are corresponding angles, and e equals f, because they are vertically opposite, so therefore a equals f." Stuff like that. Here are some of the most basic "known truths" which you might use to prove stuff. (Okay, I think they're called "axioms" or something like that, but whatever.)
Before I get started, I was really curious to find out how people know about this blog apart from the people I've told, because I would have thought you'd have to be pretty specific in your searches to get this particular blog in amongst all of the other websites displaying the same sort of information. One thing that came up was this website here: http://library.onlinepatashala.com/SubPlay.aspx?Video=admin@2509. This blog is listed under the list of blogs in the bottom right corner. Seems a bit odd that a blog like this, written in such an obnoxious style, can be listed on an educational website like that. Maybe I should make a poll or something to find out how everyone's finding out about this blog because I am one of those annoying overly curious people.
Anyway, time to get started on speed reviewing chapters 3 to 6!
Basics of 3D Vectors
3D vectors are pretty similar to 2D vectors, apart from the fact that they're obviously in 3D and as such have an extra component for the extra dimension. Instead of good ol' ai + bj, now you're dealing with ai + bj + ck. Pretty much everything else is the same, though- adding, subtracting and finding the magnitude are all the same. Yes, even in 3D you're finding sqrt(a^2 + b^2 + c^2). The procedure for working out the unit vector is also the same, as is the procedure for finding the angle between two lines. (Getting bored of this now, are you?) Yes, I need to write a blog post on how to do all this stuff, but as I said before, this is a brief review so I'm assuming that you know stuff like that from last year's Spec work.
(By the way, check out Sadler's example at the beginning of the section, particularly the guy called Captain Over. "Hi Victor, it's Over, over?" Almost as bad as Rosencrantz and Guildenstern- "I think we're off course." "Of course!" I'm not sure if that's the exact wording but I can't be bothered checking.)
Equations of Lines and Planes
The vector equation of a line in 3D works exactly the same as the equation of a line in 2D. Basically, you have the position vector of one point in the line plus a scalar multiple of the direction of the line, in the form r = a + (lambda)b.
The normal equation of a line in 3D unfortunately doesn't work the same way, as every line has multiple possible lines that are perpendicular to it. The way I helped myself visualise this one was by sticking my arm out and then holding my ruler perpendicular to my arm, and then seeing for myself that there are multiple ways in which this could be achieved. Maybe I'm just weird, or maybe it might help you visualise it too.
However, you can have a normal equation of a plane. This is because a plane (a.k.a. a big flat thing that extends in all directions, NOT the aeroplanes that fly, unfortunately) only has one direction that's perpendicular to it. Therefore, you can use the equation r (dot) n = c where c = a (dot) n, where a is a point on the plane and n is a vector perpendicular to the plane. To find out a vector perpendicular to a plane, you can do several things.
1) Sometimes you might get lucky and they'll say that the plane is perpendicular to a certain vector/plane/whatever. Then you can just use the vector normal that they give you!
2) If you get a vector parallel to the plane in question, then you need to find a vector perpendicular to that vector (i.e. if it's perpendicular to a plane parallel to your plane, then it's also perpendicular to your plane. You can see this in my post about Scalar Product and Closest Approach). To do this, you can use dot product and guess and check. I normally let the vector normal be equal to xi + yj + zk and then set the dot product of that and the parallel vector equal to 0. Then I let one of the variables be equal to 0 and then work out what y is in relation to z (for example, y = z or 2x = z). Then I pick numbers that suit that criteria (e.g. for y = z I might choose to make x = 0, y = 1 and z = 1, and for 2x = z I might make it x = 1, y = 0 and z = 2) and double check to make sure that I've wound up with a vector normal to the plane.
3) If you get given two lines, you need to find a vector normal to both lines. Find the dot product between each line and xi + yj + zk, and you'll wind up with two equations for x, y and z. This isn't enough to give you exact numbers, but it is enough to give you the relationship between the variables if you tweak your equations around a bit and/or set one of the variables equal to 0.
4) If you get given three points, then you need to find two lines by finding the position of one point relative to the other two. Then, just use the methods mentioned above.
By the way, there are also vector and Cartesian equations of a plane. The vector equation is in the form r = a + (lambda)b + (mu)c, where b and c are non-parallel vectors that are parallel to the plane. The Cartesian form is simply in the form ax + by + cz = (constant). One great thing about this form is that ai + bj + ck is a vector perpendicular to whatever plane you're looking at.
Interception between lines and planes are worked out in the same way as 2D vectors.
Quick Notes on Proofs by Deduction
"Proof by Deduction" is basically the fancy term for most of the proofs that you've probably been doing so far- you know, the ones where you say that "this equals this, because of that axiom" or whatever and you keep putting down logical facts and explanations until you prove whatever it is that had to be proved? Okay, terrible explanation. Maybe I should provide an example: trig proofs. Proofs where you say "a equals e, because they are corresponding angles, and e equals f, because they are vertically opposite, so therefore a equals f." Stuff like that. Here are some of the most basic "known truths" which you might use to prove stuff. (Okay, I think they're called "axioms" or something like that, but whatever.)
- A straight line has an angle of 180 degrees.
- Angles around a point add up to 360 degrees.
- Alternate angles are equal.
- Corresponding angles are equal.
- Co-interior angles are supplementary (add to 180 degrees).
- Angles of a triangle add to 180 degrees.
- Angles of a quadrilateral add to 360 degrees.
- Angles of an n-sided polygon add to (n - 2)*180 degrees.
- The exterior angle of a triangle equals the sum of the two interior opposite angles.
- Vertically opposite angles are equal.
There's also the counter-example method, which is where you prove or disprove something by providing a counter-example to the given statement. This is a rather simple example but if you wanted to prove that not all odd numbers are divisible by 3 you could just say that 25 isn't divisible by 3.
Proofs using vectors are also rather straightforward, except that you have to take care with the direction of the vectors as well as their magnitudes.
Proof by Exhaustion
Basically, a Proof by Exhaustion literally exhausts all of the possibilities. In Proofs by Exhaustion, you first divide all real numbers up into categories and test each category separately according to whatever you're trying to prove. For example, you might need to test all of the odd numbers and all of the even numbers, or all of the multiples of 3, all of the multiples of 3 plus one, and all of the multiples of 3 plus two. How you divide the numbers up also depends on what you're trying to prove.
Proof by Contradiction
These are NASTY. In Proofs by Contradiction, you have to prove that the inverse of whatever statement you're given isn't true, and therefore the original statement must be true. For example, you might be given the lengths of three sides of a triangle and asked to prove that it's not a right-angled triangle. You would go about this by first acting as though it is a right-angled triangle, before showing that that assertion falls flat on its face, and then concluding that the original statement- that the triangle isn't right-angled- must be true. For example, if the side lengths are 8cm, 9cm and 10cm, you could show that 8^2 + 9^2 doesn't equal 10^2 and therefore doesn't conform to Pythagoras' Theorem for right-angled triangles. Therefore, this triangle can't be right-angled. That was a relatively simple example- there are much worse examples to be encountered.
Basics of Differentiating Trig Functions
As x approaches 0, (sin x/x) = 1, and ((1-cos x)/x) = 0.
Then, using first principles (can't be bothered putting in all the working out here), the derivative of sin x is cos x, and the derivative of cos x is (-sin x). On a similar note, the derivative of (-sin x) is (-cos x) and the derivative of (-cos x) is sin x.
There are three possible derivatives for y = tan x. One is (1 / (cos x)^2), another is (sec x)^2, and a third is 1 + (tan x)^2. Just use the one that suits your mood and the question that you're doing.
The product rule, quotient rule and chain rule also work here. With the chain rule, the power takes precedence, i.e. in (cos u)^a, the derivative is a(cos u)^(a-1)(-sin u)(u'). Bear in mind that the derivative of cos u is (-sin u)(u') according to the chain rule (dy/du is -sin u in this case, and then u' is another way of notating du/dx, and then dy/dx = (dy/du)(du/dx) according to the chain rule).
Implicit Differentiation
Implicit differentiation examples are too long and annoying to type up in a brief review, so I'll just tell you the concept here. Implicitly defined functions are functions that don't have one variable on one side and the other on the other side (e.g. x^2 + y^2 = 10). When differentiating these, you can either rearrange the equation (which isn't always possible) and differentiate normally, or you can use implicit differentiation. In implicit differentiation you have to differentiate every component separately in respect to one of the variables (normally x) and then factor out the (dy/dx) parts to get an equation for (dy/dx). Normally the derivative will be in terms of both x and y.
When you have to differentiate y in respect to x, you get (dy/dx). After all, (dy/dx) can be read as "the derivative of y with respect to x." When you've got something like y^3, you have to use the chain rule, so you end up with (3y^2)(dy/dx). Finally, if you have something with x and y multiplied together, you have to use the product rule and separate the x and y. (If there are constants, pair them up with x.) For example, the derivative of 3xy with respect to x is 3y + 3x(dy/dx).
Differentiating Parametric Equations
Thankfully, these aren't too hard to do. Parametric equations are when you have, for example, an equation for x in terms of t and an equation for y in terms of t. You can differentiate both with respect to t so that you end up with an equation for (dy/dt) and one for (dx/dt). Then you can use the Chain Rule to end up with an equation for (dy/dx)! As (dy/dx) = (dy/dt)(dt/dx), all you need to do is get the reciprocal of (dx/dt) (i.e. 1/(dt/dx)) and multiply it by (dy/dt) to get (dy/dx)!
(Finding the second derivative isn't so easy, but I can't remember the process for doing so.)
So that's a review of chapters 1-6 finished! Yay! When I've got some time and a lot more motivation than I do now, I might flesh out some of the topics above.
Wednesday, April 3, 2013
Very brief review of chapters 1-6 (Sadler 3CD Spec)
Well, I have a test tomorrow, and I've actually slacked off for the past week or so, so now it's time to do some power studying. (Actually I've already done some power studying, now this is just me writing up about it.)
My idea of power studying is going through all of the topics listed in a book or course outline and writing down stuff on each of the dot points or book headings. If there's something that I can't write anything for, then I revise it.
All right. Here we go... a speedy review of chapters 1-6!
Chapter 1: Polar Coordinates
Distance between two points
Just use cosine rule (i.e. c^2 = a^2 + b^2 - 2ab cos (theta)). Polar coordinates give the lengths of two sides of the triangle so a and b are all worked out for you. As for cos (theta), theta is simply the difference between the angles given in the polar coordinates. Then solve to find c. If you get stuck, just draw a diagram and it should all make sense (if it doesn't, then revise the cosine rule and/or polar coordinate basics).
Polar equations and graphs
If you have theta = (angle), then the graph is basically a line in the direction of the angle. (i.e. if the equation is theta = pi/4, then you have a line continuing at pi/4 radians, or 45 degrees, as measured anticlockwise from the positive x-axis). If it says anywhere that r doesn't have to be greater than theta, then the line extends in both directions.
If you have r = (constant), then the graph is a circle with a radius equal to the constant.
You can also get inequalities for these as well. Remember, if it's a greater than/ less than sign without the equals bit underneath, you need to draw a dotted line, not a solid line.
Oh yeah, and there are the spirals too, in the form r = k(theta), where k is a constant, r is the magnitude and theta is the angle. Normally you see these in those questions where it asks you to write the equation for the graph. Normally it helps to use the values of r at (pi/2) and/or pi to help you determine k, and, therefore, the equation.
Chapter 2: Complex Numbers
Polar form of a complex number
Okay, there's an explanation for this, and while I tend to value understanding over memorisation, this brief summary is an exception. Basically, the polar form is in the form r cis (theta) where r is the modulus (i.e. the length between the origin and the location of the number on an Argand diagram), cis is short for "cos theta plus i sin theta" (ties in with the explanation that I'm missing here) and (theta) is the argument, or the angle between the positive x-axis and the line between the origin and the location of the complex number. Normally, theta is stated in radians, and the principal argument (the one normally used) is between negative pi and pi (including positive pi, but not including negative pi).
By the way, that old familiar form of a complex number a + bi is also called the Cartesian or rectangular form of a complex number.
To convert from rectangular form to polar form, first find the modulus by using Pythagoras' theorem (i.e. modulus = sqrt(a^2 + b^2)). Then use SOHCAHTOA to find the angle/argument. Sometimes it helps to draw a diagram to make sure you get the right angle. Remember, the angle you want is the one measured from the positive x-axis.
To convert back, draw a diagram and use SOHCAHTOA or just have a couple of formulas memorised. The ones you'll want are x = r cos (theta) and y = r sin (theta). (Hopefully I've got them right there...)
Multiplying and dividing complex numbers in polar form
Believe it or not, there ARE benefits to polar form- it's much easier to multiply and divide in this form!
To multiply: multiply the moduli, add the arguments.
To divide: divide the moduli, subtract the arguments.
Then just make sure that the argument fits into the definition of a principal argument (between negative pi and positive pi- but not including negative pi). Done!
Complex numbers- regions on a diagram
For any complex number z, if you ever get the equation |z| = (constant), then the diagram is basically a circle with the centre at the origin and a radius equal to the constant. Apparently, there's a fancy word for it- the "locus" of |z| = (constant).
Then, if you have something like |z-w| = (constant), where w is another complex number, then the diagram is basically a circle with its centre at point w and a radius equal to the constant.
And then you can do all kinds of inequality stuff here too.
Okay, stuff what I said about chapters 1 to 6. It's 9.27pm, and I want to get to bed early-ish because I have orchestra tomorrow and I think that sleep will help me more than studying. So good night, everyone :)
(By the way, I found out that I got 30 page views on the 31st of March. I was kind of surprised, to be honest, because I didn't think that anyone but me ever came on here since my blog is so nerdy, and even then I haven't updated my blog in a few months. Hmm. I'm curious now.)
My idea of power studying is going through all of the topics listed in a book or course outline and writing down stuff on each of the dot points or book headings. If there's something that I can't write anything for, then I revise it.
All right. Here we go... a speedy review of chapters 1-6!
Chapter 1: Polar Coordinates
Distance between two points
Just use cosine rule (i.e. c^2 = a^2 + b^2 - 2ab cos (theta)). Polar coordinates give the lengths of two sides of the triangle so a and b are all worked out for you. As for cos (theta), theta is simply the difference between the angles given in the polar coordinates. Then solve to find c. If you get stuck, just draw a diagram and it should all make sense (if it doesn't, then revise the cosine rule and/or polar coordinate basics).
Polar equations and graphs
If you have theta = (angle), then the graph is basically a line in the direction of the angle. (i.e. if the equation is theta = pi/4, then you have a line continuing at pi/4 radians, or 45 degrees, as measured anticlockwise from the positive x-axis). If it says anywhere that r doesn't have to be greater than theta, then the line extends in both directions.
If you have r = (constant), then the graph is a circle with a radius equal to the constant.
You can also get inequalities for these as well. Remember, if it's a greater than/ less than sign without the equals bit underneath, you need to draw a dotted line, not a solid line.
Oh yeah, and there are the spirals too, in the form r = k(theta), where k is a constant, r is the magnitude and theta is the angle. Normally you see these in those questions where it asks you to write the equation for the graph. Normally it helps to use the values of r at (pi/2) and/or pi to help you determine k, and, therefore, the equation.
Chapter 2: Complex Numbers
Polar form of a complex number
Okay, there's an explanation for this, and while I tend to value understanding over memorisation, this brief summary is an exception. Basically, the polar form is in the form r cis (theta) where r is the modulus (i.e. the length between the origin and the location of the number on an Argand diagram), cis is short for "cos theta plus i sin theta" (ties in with the explanation that I'm missing here) and (theta) is the argument, or the angle between the positive x-axis and the line between the origin and the location of the complex number. Normally, theta is stated in radians, and the principal argument (the one normally used) is between negative pi and pi (including positive pi, but not including negative pi).
By the way, that old familiar form of a complex number a + bi is also called the Cartesian or rectangular form of a complex number.
To convert from rectangular form to polar form, first find the modulus by using Pythagoras' theorem (i.e. modulus = sqrt(a^2 + b^2)). Then use SOHCAHTOA to find the angle/argument. Sometimes it helps to draw a diagram to make sure you get the right angle. Remember, the angle you want is the one measured from the positive x-axis.
To convert back, draw a diagram and use SOHCAHTOA or just have a couple of formulas memorised. The ones you'll want are x = r cos (theta) and y = r sin (theta). (Hopefully I've got them right there...)
Multiplying and dividing complex numbers in polar form
Believe it or not, there ARE benefits to polar form- it's much easier to multiply and divide in this form!
To multiply: multiply the moduli, add the arguments.
To divide: divide the moduli, subtract the arguments.
Then just make sure that the argument fits into the definition of a principal argument (between negative pi and positive pi- but not including negative pi). Done!
Complex numbers- regions on a diagram
For any complex number z, if you ever get the equation |z| = (constant), then the diagram is basically a circle with the centre at the origin and a radius equal to the constant. Apparently, there's a fancy word for it- the "locus" of |z| = (constant).
Then, if you have something like |z-w| = (constant), where w is another complex number, then the diagram is basically a circle with its centre at point w and a radius equal to the constant.
And then you can do all kinds of inequality stuff here too.
Okay, stuff what I said about chapters 1 to 6. It's 9.27pm, and I want to get to bed early-ish because I have orchestra tomorrow and I think that sleep will help me more than studying. So good night, everyone :)
(By the way, I found out that I got 30 page views on the 31st of March. I was kind of surprised, to be honest, because I didn't think that anyone but me ever came on here since my blog is so nerdy, and even then I haven't updated my blog in a few months. Hmm. I'm curious now.)
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