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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.

Do anyone knows is answers? Just the answers will be fine, but with steps is even better if you are convinient.
Many thanks.
summerleei is offline  
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,

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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