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December 1st, 2013, 07:11 AM   #1
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Some multivariable calculus problems

Here are some problems of multivariable calculus.

https://www.dropbox.com/sh/vxpjv8wejpy0fb3/XJ3ulE7m85

Do anyone knows is answers? Just the answers will be fine, but with steps is even better if you are convinient.
Many thanks.
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December 5th, 2013, 05:49 AM   #2
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Re: Some multivariable calculus problems

Summerleei, I hope you have the actual solutions to correct me where I have gone amiss. Here are my results

1.) The force field here is conservative, giving us that

2a.) For the ellipse,
b.)

3a.) has the potential function , I don't think has any potential function, and has the potential function .
b.) and
Sorry, I was lazy to compute that for G because in applying Cauchy residues theorem, the denominator has double poles.
c.) Green's Theorem may only be applied to the field F because the other two vector fields have singularities in the domain of integration.

4.) I didn't use Green's theorem for this one. (Can you please show me how to?) My result was .

5.) The flux is given by the formula
Maximal flux occurs when . Maximal flux = 37632

6.) Flux equals . Flux from the bottom and top is zero because the gauss map to the object is orthogonal to the vector field on those portions.
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