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Hi, I'm trying to compute the gradient tensor of a vector field and I must say I'm quite confused. In other words I have a vector field which is given in spherical coordinates as:

[tex]\vec{F}=\begin{bmatrix} 0 \\ \frac{1}{\sin\theta}A \\ -B \end{bmatrix} [/tex], with A,B some scalar potentials and I want to compute the gradient of the vector.

I found that I have to use the covariant derivative which is given by: [tex]\nabla\vec{B}= \frac{\partial{B_p}}{\partial{u^q}} - B_k\Gamma_{pq}^{k}[/tex]

For some reason I can't get the right results. For instance the tensor element with p=2, q=1 is given as [tex]\frac{1}{\sin\theta}\frac{\partial A}{\partial r} = \frac{\partial F_\theta}{\partial r}[/tex] whereas according the covariant derivative formula should be: [tex]\frac{\partial F_\theta}{\partial r} - \frac{1}{r} F_\theta[/tex].

Probably I'm doing a big mistake and I think I haven't really understood how I can calculate the gradient tensor. Any help will be appriciated.

[tex]\vec{F}=\begin{bmatrix} 0 \\ \frac{1}{\sin\theta}A \\ -B \end{bmatrix} [/tex], with A,B some scalar potentials and I want to compute the gradient of the vector.

I found that I have to use the covariant derivative which is given by: [tex]\nabla\vec{B}= \frac{\partial{B_p}}{\partial{u^q}} - B_k\Gamma_{pq}^{k}[/tex]

For some reason I can't get the right results. For instance the tensor element with p=2, q=1 is given as [tex]\frac{1}{\sin\theta}\frac{\partial A}{\partial r} = \frac{\partial F_\theta}{\partial r}[/tex] whereas according the covariant derivative formula should be: [tex]\frac{\partial F_\theta}{\partial r} - \frac{1}{r} F_\theta[/tex].

Probably I'm doing a big mistake and I think I haven't really understood how I can calculate the gradient tensor. Any help will be appriciated.

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