Maximize Flux: Vector Field (4x+2x^3z)i-y(x^2 +y^2)j -(3x^2z^2 +4y^2z)k

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Given a vector field (4x+2x^3z)i-y(x^2 +y^2)j -(3x^2z^2 +4y^2z)k which closed surface has the greatest flux. I imagine that the divergence theorem palys a role but I'm not sure. please anwer ! This is killing me!
 
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Since the problem is asking for a function (surface) that gives a maximum, this looks like a "calculus of variations" problem. What course is this in?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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