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         FORMULAS INVOLVING THE DEL OPERATOR

Formulas involving the del operator. Let A(x, y, z) and B(x, y, z) be differentiable vector functions and let Φ(x, y, z) and ψ(x, y, z) be differentiable scalar functions of position (x, y, z). Then

                        or        grad (Φ + ψ) = grad Φ + grad ψ

                        or        grad (Φψ) = Φ grad ψ + ψ grad Φ

3.          ∙ (A + B) = ∙ A + ∙ B                                 or      div (A + B) = div A + div B

4.          (A + B) = A + B                                      or      curl (A + B) = curl A + curl B

5.          ∙ (ΦA) = ( Φ) ∙ A + Φ( ∙ A)

6.          (ΦA) = ( Φ) A + Φ( A)

7.          ∙ (A B) = B ∙ ( A) + A ∙ ( B)

8.          (A B) = (B ∙ ) A - B( ∙ A) - (A ∙ )B + A( ∙ B)

9.          (A ∙ B) = (B ∙ ) A + ( A ∙ )B + B ( A) + A ( B)

11.        ( Φ) = 0 .                                 The curl of the gradient of Φ is zero.

12.        ∙ ( A) = 0 .                            The divergence of the curl of A is zero.

Φ and A are assumed to have continuous partial derivatives in formulas 10 - 13.                   

Legitimate (meaningful) combinations.

Reference.

  Spiegel. Vector Analysis. p. 58