## Gradient of a Function

Introduction In this post, I find an expression for the gradient of a function, in terms of spherical coordinates. This is a continuation of previous posts, such as this one. This post has a lot of symbols, but there is a lot of repetition. Formalism Recall that $\vec{e}_r = \frac{\partial x(r)}{\partial r}\big|_{r^+} \vec{e}_x + \frac{\partial y(r)}{\partial r}\big|_{r^+} \vec{e}_y + \frac{\partial z(r)}{\partial r}\big|_{r^+} \vec{e}_z$ $\vec{e}_{\theta} = \frac{1}{r} \frac{\partial x(\theta)}{\partial\theta} \big|_{\theta^+} \vec{e}_x + \frac{1}{r} \frac{\partial y(\theta)}{\partial \theta} \big|_{\theta^+} \vec{e}_y + \frac{1}{r} \frac{\partial z(\theta)}{\partial\theta} \big|_{\theta^+} \vec{e}_z$ $\vec{e}_{\phi} = \frac{1}{r \sin\theta} \frac{\partial x(\phi)}{\partial\phi}\big|_{\phi^+} \vec{e}_x + \frac{1}{r \sin\theta} \frac{\partial y(\phi)}{\partial\phi}\big|_{\phi^+} \vec{e}_{y}$. Multiply the first equation by $\partial r|_{r^+}$, multiply the second equation by […]