How te expand [tex] \nabla f \cdot (p-p_0) [/tex]in spherical polar coordinates

[zheng]

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

in spherical polar coordinates:
$$\nabla f$$ = $$\frac{\partial f}{\partial r} e_r$$+ $$\frac{1}{r sin\theta}\frac{\partial f}{\partial \phi} e_{\phi}$$+ $$\frac{1}{r}\frac{\partial f}{\partial \theta} e_{\theta}$$

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

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

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

 

[tiny-tim]

welcome to pf!

hi zheng! welcome to pf!

we're on the unit sphere, the difference in longitude is (θ - θ₀) and the difference in latitude is (φ - φ₀)

but you get more longitude than latitude for the same length, by a factor of sinθ