Final answer:
The curl of a curl of a vector field represents the rotation of the curl of the vector field. It can be expressed as ∇ × (∇ × V) using vector calculus identities. Applying these identities, we can expand and simplify the expression to ∇(∇ · V).
Step-by-step explanation:
The curl of a curl of a vector field is also known as the double curl or the Laplacian of a vector field. It represents the rotation of the curl of a vector field. Mathematically, it can be expressed as the curl of the curl of a vector field V, denoted as ∇ × (∇ × V).
To prove this, we can expand the expression using vector calculus identities and properties. Applying the identity ∇ × (∇ × V) = ∇(∇ · V) - ∇²V, we can simplify the expression to ∇(∇ · V) - ∇²V. The Laplacian of a vector field (∇²V) can be further expanded as the divergence of the gradient of V (∇ · (∇V)) - ∇²V. Then, using the vector identity ∇ · (∇V) = 0, we can simplify it to ∇(∇ · V).