Final answer:
Proving the vector calculus identity ∇×(∇×V) = ∇(∇⋅V) − ∇²V involves using properties of cross products and dot products to algebraically manipulate the vector field V and verifying the identity through vector calculus.
Step-by-step explanation:
The question asks to show that ∇×(∇×V) = ∇(∇⋅V) − ∇²V where V is a vector field. This identity is one of the vector calculus identities, which relates the curl of the curl of a vector field to the gradient of the divergence of the vector field minus the Laplacian of the vector field.
To prove this, we use vector calculus and properties of cross products and dot products in three dimensions. Let's assume a vector field V represented as V = VxÎ + VyÏ + Vz‰. We would need to calculate the curl of V, the divergence of V, and then use the results to verify the identity through algebraic manipulation involving partial derivatives and the properties of the vector operations.
Throughout the calculation, we would take advantage of the distributive property and the fact that the magnitudes of the orthogonal unit vectors (Î, Ï, and ‰) are one, as well as using the results of cross products involving these unit vectors. The proof would also involve applying the definition of the dot product to decompose the vector V into its components.
Ultimately, you would arrive at the equality that ∇×(∇×V) indeed equals ∇(∇⋅V) − ∇²V, showing how the vector calculus identity holds.