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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) = diijkAjBk)
         = εijkdi(AjBk)
         = εijk(BkdiAj + AjdiBk)
         = BkkijdiAj) - AjjikdiBk)
         = B.(curl(A)) - A.(curl(B)).

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