The Laplacian operator, denoted ∇2, is the second order partial differential operator de...

The Laplacian operator, denoted ∇², is the second order partial differential operator defined by

(a) Explain why it makes sense to think of ∇² as∇ ·∇.

(b) Show that if f and g are functions of class C², then ∇²( f g) = f ∇²g + g∇² f + 2(∇ f ·∇g).

(c) Show that ∇ · ( f ∇g − g∇ f ) = f ∇²g − g∇² f.