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!
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 $\displaystyle f(x)=\cos(2\pi x)$.
The answer to 3. is given by the hairy ball theorem.
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.
How about $\displaystyle f(x,y)=\cos(2\pi x)+\sin(2\pi y)$, for example?
Yes.
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.
That makes no kind of sense to me. The function that I suggested was $\displaystyle f(x,y)=\cos(2\pi x)+\sin(2\pi y)$, and this was meant to be an answer to problem 2(a). It is a non-constant, continuous function, with $\displaystyle f(x,0) = \cos(2\pi x) = f(x,1)$ and $\displaystyle f(0,y) = 1 + \sin(2\pi y)$ = 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.