Curl of a×b

[Reading: a is a constant vector (this is the case in which the printed result is true), and the term (a · ∇)b means the operator a₁∂/∂x + a₂∂/∂y + a₃∂/∂z applied to b. For a variable a the full identity is curl(a×b) = a div b − b div a + (b·∇)a − (a·∇)b, which reduces to the printed one when a is constant.]

Let a = (a₁, a₂, a₃) be constant and b = (b₁, b₂, b₃), functions of x, y, z.

a × b = (a₂b₃ − a₃b₂) i + (a₃b₁ − a₁b₃) j + (a₁b₂ − a₂b₁) k.

x-component of the curl

[curl(a×b)]ₓ = ∂/∂y (a₁b₂ − a₂b₁) − ∂/∂z (a₃b₁ − a₁b₃)
             = a₁ ∂b₂/∂y − a₂ ∂b₁/∂y − a₃ ∂b₁/∂z + a₁ ∂b₃/∂z

Add and subtract a₁ ∂b₁/∂x:

= a₁ (∂b₁/∂x + ∂b₂/∂y + ∂b₃/∂z) − (a₁ ∂/∂x + a₂ ∂/∂y + a₃ ∂/∂z) b₁
= a₁ (div b) − (a · ∇) b₁

Other components

The y and z components follow by cyclic change of (x, y, z) and (1, 2, 3):

[curl(a×b)]_y = a₂ div b − (a·∇)b₂
[curl(a×b)]_z = a₃ div b − (a·∇)b₃

Hence curl(a × b) = a div b − (a · ∇)b, for constant a.