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Triple Vector Products

As in vector there are two types of triple vector products that compare with the scalar triple product (that gives the area of a parallelopiped) and the vector triple product. There are differences due to the operator nature of $\nabla $. So what about $\nabla
\cdot ({\bf A} \times {\bf B})$? Remember the vector rule

\begin{displaymath}
{\bf a}\cdot ({\bf b}\times {\bf c}) = {\bf b}\cdot ({\bf c}\times
{\bf a}) = {\bf c}\cdot ({\bf a}\times {\bf b}).
\end{displaymath}

Using operator rules

\begin{displaymath}
\nabla \cdot ({\bf A} \times {\bf B}) \qquad B \nabla A + A \nabla B.
\end{displaymath}

Now including the vector rules

\begin{displaymath}
{\bf B}\cdot (\nabla \times {\bf A}) + {\bf A}\cdot ({\bf B} \times
\nabla)
\end{displaymath}

but in the second term the operator must act on ${\bf B}$. So we reverse the order of the last term to give
\begin{displaymath}
\nabla \cdot ({\bf A} \times {\bf B}) = {\bf B}\cdot \nabla \times
{\bf A}- {\bf A}\cdot \nabla \times {\bf B}.
\end{displaymath} (1.26)

An important rule is similar to the vector triple product, namely $\nabla \times ({\bf A} \times {\bf B})$. This rule will be covered in a tutorial question but we use the vector rule
\begin{displaymath}
{\bf a}\times ({\bf b} \times {\bf c}) = ({\bf a}\cdot {\bf c}){\bf b} - ({\bf a}\cdot {\bf b}){\bf c}.
\end{displaymath} (1.27)



Prof. Alan Hood
2000-11-06