Algebraic Identities
\begin{flalign}
&{\bf A}\cdot\left({\bf B}\times{\bf C}\right) = {\bf B}\cdot\left({\bf C}\times{\bf A}\right) = {\bf C}\cdot\left({\bf A}\times{\bf B}\right) & \tag{B.18} \\
&{\bf A}\times\left({\bf B}\times{\bf C}\right) = {\bf B}\left({\bf A}\cdot{\bf C}\right) - {\bf C}\left({\bf A}\cdot{\bf B}\right) & \tag{B.19}\end{flalign}
Identities Involving Differential Operators
\begin{flalign}
&\nabla \cdot \left(\nabla \times A\right) = 0 & \tag{B.20} \\
&\nabla \times \left(\nabla f\right) = 0 & \tag{B.21} \\
&\nabla \times \left(f{\bf A}\right) = f\left(\nabla\times{\bf A}\right) + \left(\nabla f\right)\times{\bf A} & \tag{B.22} \\
&\nabla \cdot \left({\bf A}\times {\bf B}\right) = {\bf B}\cdot\left(\nabla \times{\bf A}\right)-{\bf A}\cdot\left(\nabla \times{\bf B}\right) & \tag{B.23} \\
&\nabla \cdot \left(\nabla f\right) = \nabla^2 f & \tag{B.24} \\
&\nabla \times \nabla \times {\bf A} = \nabla\left(\nabla\cdot{\bf A}\right)-\nabla^2{\bf A} & \tag{B.25} \\
&\nabla^2 {\bf A} = \nabla\left(\nabla\cdot{\bf A}\right) - \nabla\times\left(\nabla\times{\bf A}\right) & \tag{B.26}\end{flalign}
Divergence Theorem: Given a closed surface \({\mathcal S}\) enclosing a contiguous volume \({\mathcal V}\),
\begin{equation}
\int_{\mathcal V} \left( \nabla \cdot {\bf A} \right) dv = \oint_{\mathcal S} {\bf A}\cdot d{\bf s} \tag{B.27}\end{equation}
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 \({\mathcal S}\),
\begin{equation}
\int_{\mathcal S} \left( \nabla \times {\bf A} \right) \cdot d{\bf s} = \oint_{\mathcal C} {\bf A}\cdot d{\bf l} \tag{B.28}\end{equation}
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.”