My Math Forum Some multivariable calculus problems

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 December 1st, 2013, 07:11 AM #1 Newbie   Joined: Dec 2013 Posts: 1 Thanks: 0 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.
 December 5th, 2013, 05:49 AM #2 Senior Member   Joined: Dec 2012 Posts: 372 Thanks: 2 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 $\int_C \overline{F}.\overline{dr}= \int_{RS} \overline{F}.\overline{dr} = 12$ 2a.) For the ellipse, $\dfrac{1}{2}\int_{\partial D} -ydx + xdy= 180\pi$ b.) $\dfrac{1}{2}\int_{\partial D} -ydx + xdy= \dfrac{147 \pi}{8}$ 3a.) $\overline{F}$ has the potential function $f(x, y)= 4xy$, I don't think $\overline{G}$ has any potential function, and $\overline{H}$ has the potential function $h(x, y)= (x^2 + y^2)^{\frac{1}{2}}$. b.)$\int_C \overline{F}.\overline{dr}= 0 \ , \ \int_C \overline{G}.\overline{dr} = -5\displaystyle{\int}_{[0, 2\pi]}\dfrac{dt}{1 + 24cos^2t}$ and $\int_C \overline{H}.\overline{dr}= 0$ 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 $\int_C \overline{F}.\overline{dr}= 18\pi$. 5.) The flux is given by the formula $bc(48a - 3a^2)$ Maximal flux occurs when $a= 8, \ b = c = 14$. Maximal flux = 37632 6.) Flux equals $24\pi$. 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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