Vector Identities

Triple Products

\[\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = \mathbf{B} \cdot (\mathbf{C} \times \mathbf{A}) = \mathbf{C} \cdot (\mathbf{A} \times \mathbf{B})\] \[\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B})\]

Product Rules

\[\begin{aligned} \nabla (fg) &= f (\nabla g) + g (\nabla f) \\ \\ \nabla (\mathbf{A} \cdot \mathbf{B}) &= \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A}) + (\mathbf{A} \cdot \nabla) \mathbf{B} + (\mathbf{B} \cdot \nabla) \mathbf{A} \\ \\ \nabla \cdot (f \mathbf{A}) &= f (\nabla \cdot \mathbf{A}) + \mathbf{A} \cdot (\nabla f) \\ \\ \nabla \cdot (\mathbf{A} \times \mathbf{B}) &= \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B}) \\ \\ \nabla \times (f \mathbf{A}) &= f (\nabla \times \mathbf{A}) - \mathbf{A} \times (\nabla f) \\ \\ \nabla \times (\mathbf{A} \times \mathbf{B}) &= (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B} - \mathbf{B} (\nabla \cdot \mathbf{A}) + \mathbf{A} (\nabla \cdot \mathbf{B}) \end{aligned}\]

Second Derivatives

\[\begin{aligned} \nabla \cdot (\nabla \times \mathbf{A}) &= 0 \\ \\ \nabla \times (\nabla f) &= 0 \\ \\ \nabla \times (\nabla \times \mathbf{A}) &= \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A} \end{aligned}\]

Fundamental Theorems

Gradient Theorem: \(\quad \int_a^b (\nabla f) \cdot d\mathbf{l} = f(b) - f(a)\)

Divergence Theorem: \(\quad \int (\nabla \cdot \mathbf{A}) \, d\tau = \oint \mathbf{A} \cdot d\mathbf{a}\)

Curl Theorem: \(\quad \int (\nabla \times \mathbf{A}) \cdot d\mathbf{a} = \oint \mathbf{A} \cdot d\mathbf{l}\)