- #1

- 996

- 5

## Homework Statement

Use index and comma notation to show:

\begin{equation*}\text{div }(\text{curl } \underline{\bf{v}}) = 0\end{equation*}

## Homework Equations

\begin{align*}

& \text{(1) div } \underline{\bf{v}} = v_{i,i} \\

& \text{(2) curl } \underline{\bf{v}} = \epsilon_{ijk} v_{j,i} \underline{\bf{e}}_k

\end{align*}

## The Attempt at a Solution

Substituting (2) into (1) I get:

\begin{equation*}

\text{(3) div }(\text{curl } \underline{\bf{v}}) = (\epsilon_{ijk} v_{j,i} \underline{\bf{e}}_k)_{p,p}

\end{equation*}

Don't know what to do next. I know the right hand side of (3) is correct because if I write out the individual terms the result is zero. But, we're not suppose to write out the terms. So, how does one tell that the right hand side of (3) is zero? Is there some special trick to interpret the expression?

P.S. I also have to show that [itex]\text{curl(grad }\phi)=\underline{\bf{0}}[/itex] but the question is really the same. How to interpret a (complex) index-comma expression without writing out the terms?

Last edited: