# Find potential Function

• Nov 18th 2010, 03:13 PM
hashshashin715
Find potential Function
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...
• Nov 18th 2010, 03:21 PM
hashshashin715
Sorry, r = to the norm of x
• Nov 19th 2010, 04:06 AM
HallsofIvy
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 $\displaystyle u= x^2+ a^2$.