Page 368 - Intro to Tensor Calculus

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APPENDIX C
VECTOR IDENTITIES

The following identities assume that A, B, C, D are differentiable vector functions of position while
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f, f 1 ,f 2 are differentiable scalar functions of position.

1. A · (B × C)= B · (C × A)= C · (A × B)
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2. A × (B × C)= B(A · C) − C(A · B)
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3. (A × B) · (C × D)= (A · C)(B · D) − (A · D)(B · C)
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4. A × (B × C)+ B × (C × A)+ C × (A × B)= 0
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5. (A × B) × (C × D)= B(A · C × D) − A(B · C × D)
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= C(A · B × C) − D(A · B × C)
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6. (A × B) · (B × C) × (C × A)= (A · B × C)
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7. ∇(f 1 + f 2 )= ∇f 1 + ∇f 2
8. ∇· (A + B)= ∇· A + ∇· B
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9. ∇× (A + B)= ∇× A + ∇× B
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10. ∇(fA)= (∇f) · A + f∇· A
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11. ∇(f 1 f 2 )= f 1 ∇f 2 + f 2 ∇f 1
12. ∇× (fA)=)∇f) × A + f(∇× A)
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13. ∇· (A × B)= B · (∇× A) − A · (∇× B)
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14. (A ·∇)A = ∇ − A × (∇× A)
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15. ∇(A · B)= (B ·∇)A +(A ·∇)B + B × (∇× A)+ A × (∇× B)
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16. ∇× (A × B)= (B ·∇)A − B(∇· A) − (A ·∇)B + A(∇· B)
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17. ∇· (∇f)= ∇ f
18. ∇× (∇f)= 0
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19. ∇· (∇× A)= 0
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20. ∇× (∇× A)= ∇(∇· A) −∇ A
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