Understanding Vector Calculus: Proofs and Applications

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Homework Statement



http://photos-f.ak.fbcdn.net/hphotos-ak-ash1/hs425.ash1/23519_677242553394_6313872_37501934_8229842_n.jpg

I'm unsure as to how to approach part b and c... I figured out part A but I don't know how it ties into part b. Is it saying that F is a function of little f composed r(x) at x^?


Intuitively I should sub in part a into part b but I'm not sure.
Also, for part c..when it says n=3, does it mean that it is the sum of all the functions of x from 1 to 3? so the function I'm computing the Hessian for would be 1/3x^2?
 
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For b I think you are supposed to realize that grad(g(r))=g'(r)*grad(r). I'm not really sure where they are going with c.
 

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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