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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

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Fluid
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  • You need something as $\nabla \phi = \partial_r \phi \hat{r} + \dots$? – R.W Mar 15 '17 at 02:24
  • Nope! looking for $\vec{\nabla}\vec{A}$ which is a second rank tensor – Fluid Mar 16 '17 at 05:14
  • @Fluid https://physics.stackexchange.com/questions/356630/spatial-differentiation-of-unit-vectors –  May 20 '19 at 15:27

1 Answers1

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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

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