There are many vector differential identities linking the three vector differential operators (gradient, divergence and curl) to one another. They are extremely useful in applied mathematics and theoretical physics.
Let f be any scalar function of position, and let A and B be vector functions of position. Then the following statements are true:
div(curl(A)) = 0
curl(grad(A)) = 0
div(fA) = (A.grad)f + fdiv(A)
curl(fA) = grad(f)^A + fcurl(A)
div(A^B) = B.(curl(A)) - A.(curl(B))
curl(A^B) = (B.grad)A - B(div(A)) - (A.grad)B + A(div(B))
grad(A.B) = (B.grad)A + B^(curl(A)) + (A.grad)B + A^(curl(B))
div(grad(A)) = grad(div(A)) - curl(curl(A)).
All these results can be proved using summation convention
. For example, consider the fifth expression above.
div(A^B) = di(εijkAjBk)
= εijk(BkdiAj + AjdiBk)
= Bk(εkijdiAj) - Aj(εjikdiBk)
= B.(curl(A)) - A.(curl(B)).