# B.3: Vector Identities

[m0140_Vector_Identities]

Algebraic Identities

&A(B) = B(C) = C(A) & [m0140_eVI1]
&A(B) = B(A) - C(A) [m0140_eVI2] &

Identities Involving Differential Operators

&(A) = 0 &
&(f) = 0 &
&(fA) = f() + (f) &
&(A) = B()-A() &
&(f) = ^2 f &
& = ()-^2A &
&^2 A = () - () &

Divergence Theorem: Given a closed surface $${\mathcal S}$$ enclosing a contiguous volume $${\mathcal V}$$, $\int_{\mathcal V} \left( \nabla \cdot {\bf A} \right) dv = \oint_{\mathcal S} {\bf A}\cdot d{\bf s}$ where the surface normal $$d{\bf s}$$ is pointing out of the volume.

Stokes’ Theorem: Given a closed curve $${\mathcal C}$$ bounding a contiguous surface $${\mathcal S}$$, $\int_{\mathcal S} \left( \nabla \times {\bf A} \right) \cdot d{\bf s} = \oint_{\mathcal C} {\bf A}\cdot d{\bf l}$ where the direction of the surface normal $$d{\bf s}$$ is related to the direction of integration along $${\mathcal C}$$ by the “right hand rule.”