1.02 Using index notation, show Lagrange's identity, (A x B) · (A x B) = (A · A)(B · B) - (A · B)^2
We can treat the nabla / del operator in components as: ∇ ↦ ∂^i ≡ ∂/∂x_i. Using this, the gradient, divergence, and curl can be expressed in index notation:
Gradient: (∇f)^i = ∂^if
Divergence: ∇ · v = δ_ij∂^iv^j
Curl: (∇ x v)^i = ε^i_jk∂^jv^k
1.03 Write out the Laplacian of a scalar function ∇^2f = ∇ · ∇f in index notation and then carry out the sum.
1.04 Prove that the curl of the gradient is zero: ∇ x (∇f) = 0.
1.05 Prove that the curl of the curl is given by ∇ x (∇ x A) = ∇(∇ · A) - ∇^2A.