Introduction to Electrodynamics · Problem 1.13

Vector Analysis — David J. Griffiths

Physics · gradient

Problem

Let r\boldsymbol{\scriptstyle{r}} be the separation vector from a fixed point (x′,y′,z′)(x', y', z') to the point (x,y,z)(x, y, z), and let r\scriptstyle{r} be its length. Show that (a) ∇(r2)=2r\nabla(\scriptstyle{r}^2) = 2\boldsymbol{\scriptstyle{r}}. (b) ∇(1/r)=−r^/r2\nabla(1/\scriptstyle{r}) = -\hat{\boldsymbol{\scriptstyle{r}}}/\scriptstyle{r}^2. (c) What is the general formula for ∇(rn)\nabla(\scriptstyle{r}^n)?

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