12.3: Vector Identities
- Page ID
- 24862
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Algebraic Identities
\[ \begin{align} \mathbf { A } \cdot ( \mathbf { B } \times \mathbf { C } ) &= \mathbf { B } \cdot ( \mathbf { C } \times \mathbf { A } ) = \mathbf { C } \cdot ( \mathbf { A } \times \mathbf { B } ) \\[5pt] \mathbf { A } \times ( \mathbf { B } \times \mathbf { C } ) &= \mathbf { B } ( \mathbf { A } \cdot \mathbf { C } ) - \mathbf { C } ( \mathbf { A } \cdot \mathbf { B } ) \end{align} \nonumber \]
Identities Involving Differential Operators
\[ \begin{align} \nabla \cdot ( \nabla \times \mathbf { A } ) &= 0\\[5pt] \nabla \times ( \nabla f ) &= 0\\[5pt] \nabla \times ( f \mathbf { A } ) &= f ( \nabla \times \mathbf { A } ) + ( \nabla f ) \times \mathbf { A }\\[5pt] \nabla \cdot ( \mathbf { A } \times \mathbf { B } ) &= \mathbf { B } \cdot ( \nabla \times \mathbf { A } ) - \mathbf { A } \cdot ( \nabla \times \mathbf { B } )\\[5pt] \nabla \cdot ( \nabla f ) &= \nabla ^ { 2 } f\\[5pt] \nabla \times \nabla \times \mathbf { A } &= \nabla ( \nabla \cdot \mathbf { A } ) - \nabla ^ { 2 } \mathbf { A }\\[5pt] \nabla ^ { 2 } \mathbf { A }& = \nabla ( \nabla \cdot \mathbf { A } ) - \nabla \times ( \nabla \times \mathbf { A } )\end{align} \nonumber \]
Divergence Theorem
Given a closed surface \({\mathcal S}\) enclosing a contiguous volume \({\mathcal V}\), \[\int_{\mathcal V} \left( \nabla \cdot {\bf A} \right) dv = \oint_{\mathcal S} {\bf A}\cdot d{\bf s} \nonumber \] 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 \({\mathcal S}\), \[\int_{\mathcal S} \left( \nabla \times {\bf A} \right) \cdot d{\bf s} = \oint_{\mathcal C} {\bf A}\cdot d{\bf l} \nonumber \] 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.”