Let S be the set of all x =/= O in R^n. Let r = abs(x), and let f be the vector field defined on S by the equation:
f(x) = (r^p)(x),
where p is a real constant. Find a potential function for f on S.
Need a starting push...
So what you really have is $\displaystyle ||\vec{x}||^p \vec{x}$?
In 3 dimensions, that is $\displaystyle x(x^2+ y^2+ z^2)^{p/2}\vec{i}+ y(x^2+ y^2+ z^2)^{p/2}\vec{j}+ z(x^2+ y^2+ z^2)^{p/2}\vec{k}$ and you want a function f(x, y, z) such that
$\displaystyle \frac{\partial f}{\partial x}= x(x^2+ y^2+ z^2)^{p/2}$
$\displaystyle \frac{\partial f}{\partial y}= y(x^2+ y^2+ z^2)^{p/2}$
$\displaystyle \frac{\partial f}{\partial z}= z(x^2+ y^2+ z^2)^{p/2}$
Okay, what is an anti-derivative of $\displaystyle x(x^2+ a^2)^{p/2}$?
I recommend the substitution $\displaystyle u= x^2+ a^2$.