(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider the scalar field

[tex] V = r^n , n ≠ 0[/tex]

expressed in spherical coordinates. Find it's gradient [itex]\nabla V[/itex] in

a.) cartesian coordinates

b.) spherical coordinates

2. Relevant equations

cartesian version:

[tex]\nabla V = \frac{\partial V}{\partial x}\hat{x} + \frac{\partial V}{\partial y}\hat{y} + \frac{\partial V}{\partial z}\hat{z} [/tex]

spherical version:

[tex] \nabla V = \frac{\partial V}{\partial r}\hat{r} + \frac{1}{r}*\frac{\partial V}{\partial \phi}\hat{\phi} + \frac{1}{r*sin(\phi)}*\frac{\partial V}{\partial \theta}\hat{\theta} [/tex]

conversion:

[tex] r = (x^2+y^2+z^2)^\frac{1}{2} [/tex]

3. The attempt at a solution

a.) using the third equation...

[tex] V = r^n = (x^2+y^2+z^2)^\frac{n}{2} [/tex]

using the first equation and skipping some steps involving the chain rule...

[tex] \nabla V = \frac{n(x\hat{x}+y\hat{y}+z\hat{z})}{(x^2+y^2+z^2)^\frac{n}{2}} [/tex]

b.)Using the second equation

[tex] \nabla V = nr^m \hat{r} [/tex]

[tex]m = n-1[/tex]

Those are my two solutions to this problem. Are these right? Are they wrong? If so where did I go wrong?

Thanks!

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# Homework Help: Gradient of a scalar field

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