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Engineering LibreTexts

Appendix B3: Mathematical Formulas - Vector Identities

  • Page ID
    9417
  • Algebraic Identities

    \[ \begin{align} \mathbf { A } \cdot ( \mathbf { B } \times \mathbf { C } ) &= \mathbf { B } \cdot ( \mathbf { C } \times \mathbf { A } ) = \mathbf { C } \cdot ( \mathbf { A } \times \mathbf { B } ) \\[5pt] \mathbf { A } \times ( \mathbf { B } \times \mathbf { C } ) &= \mathbf { B } ( \mathbf { A } \cdot \mathbf { C } ) - \mathbf { C } ( \mathbf { A } \cdot \mathbf { B } ) \end{align}\]

    Identities Involving Differential Operators

    \[ \begin{align} \nabla \cdot ( \nabla \times A ) &= 0\\[5pt] \nabla \times ( \nabla f ) &= 0\\[5pt] \nabla \times ( f \mathbf { A } ) &= f ( \nabla \times \mathbf { A } ) + ( \nabla f ) \times \mathbf { A }\\[5pt] \nabla \cdot ( \mathbf { A } \times \mathbf { B } ) &= \mathbf { B } \cdot ( \nabla \times \mathbf { A } ) - \mathbf { A } \cdot ( \nabla \times \mathbf { B } )\\[5pt] \nabla \cdot ( \nabla f ) &= \nabla ^ { 2 } f\\[5pt] \nabla \times \nabla \times \mathbf { A } &= \nabla ( \nabla \cdot \mathbf { A } ) - \nabla ^ { 2 } \mathbf { A }\\[5pt] \nabla ^ { 2 } \mathbf { A }& = \nabla ( \nabla \cdot \mathbf { A } ) - \nabla \times ( \nabla \times \mathbf { A } )\end{align}\]

    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.”

    Contributors