Maximizing Flux: Gauss's Theorem

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Homework Statement



Use the Divergence (Gauss's Theorem) to find the outward oriented closed surface (no boundary) for which the flux of F(x,y,z) = (16x-xz^2)i-(y^3)j-(zx^2)k is maximized.

Homework Equations



Gauss's Theorem

The Attempt at a Solution


divF = 16-z^2-3y^2-x^2 > 0 I think ?
 
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Yes. But 16-z^2-3y^2-x^2 > 0 describes a volume. What's the surface that bounds that volume?
 

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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