# Need Help With Gradient (Spherical Coordinates)

## Homework Statement

Find te gradient of the following function f(r) = rcos(##\theta##) in spherical coordinates.

## Homework Equations

\nabla f = \frac{\partial f}{\partial r} \hat{r} + (\frac{1}{r}) \frac{\partial f}{\partial \theta} \hat{\theta} + \frac{1}{rsin\theta} \frac{\partial f}{\partial \phi} \hat{\phi}

## The Attempt at a Solution

I know that z = rcos##\theta##

But I don't know where to go from there since I don't see any ##\hat{r}##, ##\hat{\theta}##, ##\hat{\phi}##

## Answers and Replies

ShayanJ
Gold Member
Do you know how to calculate ## \frac{\partial f}{\partial r} ##,## \frac{\partial f}{\partial \theta} ## and ## \frac{\partial f}{\partial \phi} ##?

Do you know how to calculate ## \frac{\partial f}{\partial r} ##,## \frac{\partial f}{\partial \theta} ## and ## \frac{\partial f}{\partial \phi} ##?

if you're asking if I know how to take partial derivatives, then yes. The issue lies in I don't know where to begin since there is ##\hat{r}##, ##\hat{\theta}##, ##\hat{\phi}## in the equation.

ShayanJ
Gold Member
Its just like gradient in the Cartesian coordinates. You calculate ## \frac{\partial f}{\partial x} ##,etc. and assume they are components of a vector field. So you have ## \vec \nabla f=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z})=\frac{\partial f}{\partial x}\hat x+\frac{\partial f}{\partial y}\hat y+\frac{\partial f}{\partial z}\hat z ##.
What is so different about spherical coordinates?

So would it be:

\begin{split}
f(r) = rcos(\theta) \\
\frac{\partial f}{\partial r} = cos\theta \\
\frac{\partial f}{\partial \theta} = - rsin \theta \\
\frac{\partial f}{\partial \phi} = 0
\end{split}

So the gradient is

cos\theta \ \hat{r} + -sin\theta \ \hat{\theta}

Is this correct?

ShayanJ
Gold Member
Yes, that's correct.

Yes, that's correct.

Thank you, very much! :)