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

$${\bf a}\cdot ({\bf b}\times {\bf c}) = {\bf b}\cdot ({\bf c}\times {\bf a}) = {\bf c}\cdot ({\bf a}\times {\bf b}).$$

Using operator rules

$$\nabla \cdot ({\bf A} \times {\bf B}) \qquad B \nabla A + A \nabla B.$$

Now including the vector rules

$${\bf B}\cdot (\nabla \times {\bf A}) + {\bf A}\cdot ({\bf B} \times \nabla)$$

but in the second term the operator must act on ${\bf B}$. So we reverse the order of the last term to give

$$\nabla \cdot ({\bf A} \times {\bf B}) = {\bf B}\cdot \nabla \times {\bf A}- {\bf A}\cdot \nabla \times {\bf B}.$$ (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

$${\bf a}\times ({\bf b} \times {\bf c}) = ({\bf a}\cdot {\bf c}){\bf b} - ({\bf a}\cdot {\bf b}){\bf c}.$$ (1.27)