So this one might be a little simple for some of you but I was hoping I could get all the nuts and bolts needed to show this for myself.
I have the following relationship, which makes use of the the material derivative:
$$
(\vec{A}\cdot{\nabla})\vec{r}=\vec{A}
$$
I am needing to show this result in spherical polar coordinates.
Now, I don't want to be vague in what I have so far, but I really have very little.
I've started with $\vec{r}$ in spherical polar coordinates being:
$$
\vec{r}=x\mathbf{\hat{e}}_x+y\mathbf{\hat{e}}_y+z\mathbf{\hat{e}}_z
$$
Firstly, is this right? It doesn't feel right. Would the following be more appropriate:
$$
\vec{r}=\sin{\theta}\cos{\phi}\mathbf{\hat{e}}_x+\sin{\theta}\sin{\phi}\mathbf{\hat{e}}_y+\cos{\theta}\mathbf{\hat{e}}_z
$$
Secondly, the $\nabla$ operator in spherical polar coordinates, I have given as:
$$
\nabla=\frac{\partial}{\partial{r}}\vec{r}+\frac{1}{r}\frac{\partial}{\partial{\theta}}\vec{\theta}+\frac{1}{r\sin{\theta}}\frac{\partial}{\partial{\phi}}\vec{\phi}
$$
I'm just lost as to whether this is right, and if it is right how to put it all together…any nudges in the right direction would be greatly appreciated.
Thank you.
Addendum for Lydia:
$$
\frac{\partial{\vec{r}}}{\partial{R}}=\sin\theta\cos\phi\mathbf{i}+\sin\theta\sin\phi\mathbf{j}+\cos\theta\mathbf{k}
$$
But, how would I evaluate:
$$
\left|\left|{\frac{\partial{\vec{r}}}{\partial{R}}}\right|\right|
$$
Best Answer
Here are some nudges. I will use $R$ for the radius in spherical coordinates to avoid any confusion with $\vec{r}$, the position vector.
You can write $\vec{r}=R\sin\theta\cos\phi\boldsymbol{i}+R\sin\theta\sin\phi\boldsymbol{j}+R\cos\theta\boldsymbol{k}$.
Let $\{\boldsymbol{e}_R,\boldsymbol{e}_\theta, \boldsymbol{e}_\phi\}$ be the orthonormal basis vectors for spherical coordinates. Then $$\boldsymbol{e}_R=\frac{1}{\left\|\frac{\partial\vec{r}}{\partial R}\right\|}\frac{\partial\vec{r}}{\partial R}, \qquad \boldsymbol{e}_\theta=\frac{1}{\left\|\frac{\partial\vec{r}}{\partial \theta}\right\|}\frac{\partial\vec{r}}{\partial \theta}, \qquad \boldsymbol{e}_\phi=\frac{1}{\left\|\frac{\partial\vec{r}}{\partial \phi}\right\|}\frac{\partial\vec{r}}{\partial \phi}. $$
Using 1, you can calculate the above basis vectors in terms of $\boldsymbol{i},\boldsymbol{j},\boldsymbol{k}$. You should get $$\boldsymbol{e}_\phi =-\sin\phi\boldsymbol{i}+\cos\phi\boldsymbol{j}, \quad \boldsymbol{e}_\theta =\cos\theta\cos\phi\boldsymbol{i} + \cos\theta\sin\phi\boldsymbol{j}-\sin\theta\boldsymbol{k}, \quad \boldsymbol{e}_R =\sin\theta\cos\phi\boldsymbol{i} + \sin\theta\sin\phi\boldsymbol{j}+\cos\theta\boldsymbol{k}.$$ Another hint is that you should have $$\vec{r}=R\boldsymbol{e}_{R}.$$
Since the Cartesian and spherical forms of $\vec{A}$ must be equivalent, $$A_x\boldsymbol{i}+A_y\boldsymbol{j}+A_z\boldsymbol{k}=A_R\boldsymbol{e}_R+A_\theta\boldsymbol{e}_\theta+A_\phi\boldsymbol{e}_\phi.$$ Taking the dot product of both sides with $\boldsymbol{e}_R$ gives that $$A_R = A_x\boldsymbol{i}\cdot\boldsymbol{e}_R+A_y\boldsymbol{j}\cdot\boldsymbol{e}_R+A_z\boldsymbol{k}\cdot\boldsymbol{e}_R.$$ Use part 2 to evaluate $\boldsymbol{i}\cdot\boldsymbol{e}_R$, $\boldsymbol{j}\cdot\boldsymbol{e}_R$, and $\boldsymbol{k}\cdot\boldsymbol{e}_R$. You should find that $$A_R=\sin\theta\cos\phi A_x+\sin\theta\sin\phi A_y+\cos\theta A_z.$$ Repeat the above process for $\boldsymbol{e}_\theta$ and $\boldsymbol{e}_\phi$. You will now have the components of $\vec{A}$ in the spherical basis.
You can now plug everything in. Note that $$\nabla=\frac{\partial}{\partial R}\boldsymbol{e}_R+\frac{1}{R}\frac{\partial}{\partial\theta}\boldsymbol{e}_\theta+\frac{1}{R\sin\theta}\frac{\partial}{\partial\phi}\boldsymbol{e}_\phi.$$