I have a problem in my textbook, the context is not very important but it does have the equation
$[c \cdot (b \cdot \nabla)-b \cdot (c \cdot \nabla)]a = (\nabla \times a)\cdot(b \times c)$
a, b, c are all vector fields. My question is whether or not the LHS of the equation is proper notation or not. Obviously the answer is a scalar so can I expand the LHS and write
$(c \cdot a)(b \cdot \nabla)-(b \cdot a)(c \cdot \nabla) = (\nabla \times a)\cdot(b \times c)$
Why/why not?
The identity should be $$(\underline{\nabla}\times\underline{a})\cdot(\underline{b}\times\underline{c})=(\underline{\nabla}\cdot\underline{b})(\underline{a}\cdot\underline{c})-(\underline{a}\cdot\underline{b})(\underline{\nabla}\cdot\underline{c})$$