Question

Show that for two scalar fields f and g:
∇.[∇f ×( f ∇g)]= 0.

Answer 1

$$\vec{∇}*[\vec{∇}f\times f\vec{∇}g]=[\vec{∇}\times\vec{∇}f] *f\vec{∇}g=0$$

As

$$[\vec{∇}\times\vec{∇}f]=curl(grad(f))=0$$

We also used the property of triple product:

$$\vec{a}*[\vec{b}\times\vec{c}]=[\vec{a}\times\vec{b}]*\vec{c}$$