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FORMULAS INVOLVING THE DEL OPERATOR

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

1.         ∇(Φ + ψ) = ∇Φ + ∇ψ                     or        grad (Φ + ψ) = grad Φ + grad ψ

2.         ∇Φψ = Φ∇ψ + ψ∇Φ                        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.         ∇(AB) = (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.

13.       ∇×(∇×A) = ∇(∇∙A) - ∇2A

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

Legitimate (meaningful) combinations.

Reference.

Spiegel. Vector Analysis. p. 58