Gradient of a vector in spherical coordinates Where can I find the gradient of a vector field in spherical coordinates?
Wikipedia has it for cylindrical coordinates:
https://en.wikipedia.org/wiki/Tensors_in_curvilinear_coordinates#Example:_Cylindrical_polar_coordinates
I need it for spherical coordinates.
In the case of duplicate question assumption, This is NOT my answer:
Gradient of vector field in spherical coordinates
 A: You can find it in reference 1 (page 52).
For spherical coordinates $(r, \phi, \theta)$, given by 
$$x = r\sin\phi\cos\theta\, ,\quad y = r\sin\phi\sin\theta\, ,\quad z = r\cos\phi\, .$$
The gradient (of a vector) is given by
$$\nabla \mathbf{A} = \frac{\partial A_r}{\partial r} \hat{\mathbf{e}}_r \hat{\mathbf{e}}_r + \frac{\partial A_\phi}{\partial r} \hat{\mathbf{e}}_r \hat{\mathbf{e}}_\phi + \frac{1}{r}\left(\frac{\partial A_r}{\partial \phi} - A_\phi\right)\hat{\mathbf{e}}_\phi \hat{\mathbf{e}}_r + \frac{\partial A_\theta}{\partial r} \hat{\mathbf{e}}_r \hat{\mathbf{e}}_\theta
+ \frac{1}{r\sin\phi}\left(\frac{\partial A_r}{\partial \theta} - A_\theta\sin\phi\right)\hat{\mathbf{e}}_\theta \hat{\mathbf{e}}_r  + \frac{1}{r}\left(A_r + \frac{\partial A_\phi}{\partial \phi}\right)\hat{\mathbf{e}}_\phi \hat{\mathbf{e}}_\phi + \frac{1}{r}\frac{\partial A_\theta}{\partial \phi}\hat{\mathbf{e}}_\phi \hat{\mathbf{e}}_\theta
+ \frac{1}{r\sin\phi}\left(\frac{\partial A_\phi}{\partial \theta} - A_\theta\cos\phi\right)\hat{\mathbf{e}}_\theta \hat{\mathbf{e}}_\phi + \frac{1}{r\sin\phi}\left(A_r \sin\phi + A_\phi\cos\phi + \frac{\partial A_\theta}{\partial \theta}\right)\hat{\mathbf{e}}_\theta \hat{\mathbf{e}}_\theta  \, ,$$
or, in matrix notation
$$\nabla \mathbf{A} = 
\begin{bmatrix}
\frac{\partial A_r}{\partial r} &\frac{\partial A_\phi}{\partial r} &\frac{\partial A_\theta}{\partial r}\\
\frac{1}{r}\left(\frac{\partial A_r}{\partial \phi} - A_\phi\right) &\frac{1}{r}\left(A_r + \frac{\partial A_\phi}{\partial \phi}\right) &\frac{1}{r}\frac{\partial A_\theta}{\partial \phi}\\
\frac{1}{r\sin\phi}\left(\frac{\partial A_r}{\partial\theta} - A_\theta\sin\phi\right)
&\frac{1}{r\sin\phi}\left(\frac{\partial A_\phi}{\partial\theta} - A_\theta\cos\phi\right)
&\frac{1}{r\sin\phi}\left(A_r\sin\phi + A_\phi\cos\phi + \frac{\partial A_\theta}{\partial\theta}\right)\, .
\end{bmatrix}
$$
Reference


*

*J.N. Reddy  (2013). An Introduction to Continuum Mechanics, 2nd edition, Cambridge University Press.

