Question

Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If f is a scalar field and F, G are vector fields, then fF, F. G, and FXG are defined by

(f F)(x, y, z) = f(x, y, z) F(x, y, z)
(F ⋅ G)(x, y, z) = F(x, y, z) ⋅ G(x, y, z)
(F ✖ G)(x, y, z) = F(x, y, z) ✖ G(x, y, z)

div(∇f✖∇g) = 0

Answer 1

To prove the identity div(∇f✖∇g) = 0, we can expand the expression ∇✖(∇f✖∇g) using the vector triple product and apply the given assumptions of (∇⋅∇f) = 0 and (∇⋅∇g) = 0 to simplify the expression. This ultimately leads to the conclusion that div(∇f✖∇g) = 0.

Step-by-step explanation:

To prove the identity, we need to show that div(∇f✖∇g) = 0. Here's how:

1. Using the vector identity ∇✖(F✖G) = G(∇⋅F) - F(∇⋅G) + (F⋅∇)G - (G⋅∇)F, we can rewrite div(∇f✖∇g) as ∇✖(∇f✖∇g).
2. We can then expand this expression using the vector triple product:

∇✖(∇f✖∇g) = (∇g)(∇⋅∇f) - (∇f)(∇⋅∇g) + (∇⋅∇g)(∇f) - (∇⋅∇f)(∇g).

1. Since the cross product (∇g)(∇f) - (∇f)(∇g) is equal to zero, we are left with (∇⋅∇g)(∇f) - (∇⋅∇f)(∇g).

But (∇⋅∇g) = 0 and (∇⋅∇f) = 0 according to the given assumption.

Therefore, (∇⋅∇g)(∇f) - (∇⋅∇f)(∇g) = 0, proving the identity div(∇f✖∇g) = 0.