Module 1 : A Crash Course in Vectors
Lecture 4 : Gradient of a Scalar Function

Example 16 :

Divergence of $\vec r = \hat\imath x+ \hat\jmath y+\hat k z$
Divergence of position vector $\vec r$is very useful to remember.

\begin{displaymath}\nabla\cdot\vec r= \frac{\partial x}{\partial x}+ \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1+1+1=3\end{displaymath}

One can also calculate easily in spherical coordinate since $\vec r$only has radial component

\begin{displaymath}\nabla\cdot\vec r = \frac{1}{r^2}\frac{\partial}{\partial r} (r^2.r)= \frac{1}{r^2}.3r^2=3\end{displaymath}

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