vector differential identities

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)).

div(A^B) = di(εijkAjBk)
         = εijkdi(AjBk)
         = εijk(BkdiAj + AjdiBk)
         = Bk(εkijdiAj) - Aj(εjikdiBk)
         = B.(curl(A)) - A.(curl(B)).