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zheng

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**how to expand grad f * (p-p_0) in spherical polar coordinates**

in spherical polar coordinates:

[tex]\nabla f[/tex] = [tex]\frac{\partial f}{\partial r} e_r[/tex]+ [tex]\frac{1}{r sin\theta}\frac{\partial f}{\partial \phi} e_{\phi}[/tex]+ [tex]\frac{1}{r}\frac{\partial f}{\partial \theta} e_{\theta}[/tex]

[tex]p=(r,\phi,\theta)[/tex] and [tex]p_0=(r_0,{\phi}_0,{\theta}_0)[/tex] is the position vectors.

in [tex]r=r_0=1[/tex] surface, what is [tex]\left[\nabla f\right]_0 \cdot (p-p_0)[/tex], where [tex]\left[\nabla f\right]_0[/tex] is the gradient of f in position [tex]p_0[/tex]

in one paper, the answer is [tex]\left[\nabla f\right]_0 \cdot (p-p_0)=\left[\frac{1}{sin\theta}\frac{\partial f}{\partial \phi} \right]_0 \left[sin\theta (\phi-{\phi}_0)\right]+\left[ \frac{\partial f}{\partial \theta} \right]_0 (\theta-{\theta}_0)[/tex]. I do not know why the second [tex]sin\theta[/tex] is needed.

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