# B.3: Vector Identities

- Page ID
- 6379

[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 &

&(f**A**) = f() + (f) &

&(**A**) = **B**()-**A**() &

&(f) = ^2 f &

& = ()-^2**A** &

&^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.”