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Math Help - Calculus III homework

  1. #1
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    Calculus III homework

    I was given a vector calculus homework in CalculusIII, but I have no idea where to start . The professor has been saying for a few days now that it's very difficult. Any hints would be really appreciated!
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  2. #2
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    For 2., all you need is to write down a function that has period 1 in x and in y. If it was a function of one variable, you could use something like f(x)=\cos(2\pi x).

    The answer to 3. is given by the hairy ball theorem.
    Last edited by Opalg; November 1st 2007 at 01:17 AM.
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    Im kinda lost.

    2) Let's say I use f(x,y) = cos(2pi xy). When y = 0, the answer would be 1. However if y=1, i would have a variable cos(2pi x). To achieve period 1 i guess the easiest functions to use would be sin and cos, but due to the problem I just described, i dont know how to proceed.

    1) I guess here i'd just graph a function similar to the one im trying to find in 2?

    3) I guess i'd be possible, but it would never be smooth.
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  4. #4
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    Quote Originally Posted by Eleuterio View Post
    2) Let's say I use f(x,y) = cos(2pi xy). When y = 0, the answer would be 1. However if y=1, i would have a variable cos(2pi x). To achieve period 1 i guess the easiest functions to use would be sin and cos, but due to the problem I just described, i dont know how to proceed.
    How about f(x,y)=\cos(2\pi x)+\sin(2\pi y), for example?

    Quote Originally Posted by Eleuterio View Post
    1) I guess here i'd just graph a function similar to the one im trying to find in 2?
    Yes.

    Quote Originally Posted by Eleuterio View Post
    3) I guess i'd be possible, but it would never be smooth.
    That is what the hairy ball theorem says: you cannot comb a spherical cat in such a way that the hair lies flat, without any crowns or tufts.
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    I showed the professor the equation described in this post, but he said it was wrong. He wants an equation that meets the following criteria: (see attachment)


    Any help is really appreciated. I have to hand this in on wednesday!
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  6. #6
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    Quote Originally Posted by Eleuterio View Post
    I showed the professor the equation described in this post, but he said it was wrong. He wants an equation that meets the following criteria: (see attachment)
    That makes no kind of sense to me. The function that I suggested was f(x,y)=\cos(2\pi x)+\sin(2\pi y), and this was meant to be an answer to problem 2(a). It is a non-constant, continuous function, with f(x,0) = \cos(2\pi x) = f(x,1) and f(0,y) = 1 + \sin(2\pi y) = f(1,y)[/tex]. What's wrong about that? If the prof doesn't like it, then I can only assume that he was thinking in terms of a solution to problem 1. He probably wanted to see a diagram of the flow lines of a non-constant vector field on the torus. You can get this by using your answer to problem 2(a) to write down the equation for such a vector field, as indicated in problem 2(b), and then drawing a sketch of its flow lines.
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  7. #7
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    The function you gave me also makes sense to me. I really dont know what he was thinking. As for number 3, he said that with my answer (if it's correct), i should interpret that "there's always a point on Earth where the wind doesnt flow".
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