Need Help With Gradient (Spherical Coordinates)

[Dopplershift]

Homework Statement

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

Homework Equations

\begin{equation}
\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}
\end{equation}

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 $\hat{r}$, $\hat{\theta}$, $\hat{\phi}$

 

[ShayanJ]

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

 

[Dopplershift]

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]

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?

 

[Dopplershift]

So would it be:

\begin{equation}
\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}
\end{equation}
So the gradient is
\begin{equation}
cos\theta \ \hat{r} + -sin\theta \ \hat{\theta}
\end{equation}

Is this correct?

 

[ShayanJ]

Yes, that's correct.

 

[Dopplershift]

Yes, that's correct.

Thank you, very much! :)