Question

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

Answer 1

$$\nabla \cdot (A \times B)=B \cdot (\nabla \times A)-A \cdot (\nabla \times B) \space \space \space \space (1)$$ $$\nabla \times \nabla a = 0 \space \space \space \space (2)$$ $$\nabla \times (a \cdot A)=a \nabla \times A + \nabla a \times A \space \space \space \space (3)$$ $$A \cdot (B \times C)=C \cdot (A \times B) =B \cdot (C \times A) \space \space \space \space (4)$$

$$\nabla \cdot [\nabla f \times (f \cdot \nabla g)] = (1)=(f \cdot \nabla g) \cdot (\nabla \times \nabla f)- \nabla f \cdot (\nabla \times (f \cdot \nabla g))=$$ $$(2)=- \nabla f \cdot (\nabla \times (f \cdot \nabla g))=(3)=- \nabla f \cdot (f (\nabla \times \nabla g) +$$ $$\nabla f \times \nabla g )=(2)=- \nabla f \cdot (\nabla f \times \nabla g )=(4)=- \nabla g \cdot (\nabla f \times \nabla f )=0$$