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16: Appendix E- Vector Identities

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  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    The following are true for all vectors \(\vec{u}\), \(\vec{v}\), \(\vec{w}\), and \(\vec{x}\) and scalars \(\phi\) and \(\psi\) that vary continuously in space.

      Algebraic identities:
    1. \((\vec{u} \times \vec{v}) \cdot \vec{w}=(\vec{w} \times \vec{u}) \cdot \vec{v}=(\vec{v} \times \vec{w}) \cdot \vec{u}\)
    2. \(\vec{u} \times(\vec{v} \times \vec{w})=(\vec{u} \cdot \vec{w}) \vec{v}-(\vec{u} \cdot \vec{v}) \vec{w}\)
    3. \((\vec{u} \times \vec{v})(\vec{w} \times \vec{x})=(\vec{u} \cdot \vec{w})(\vec{v} \cdot \vec{x})-(\vec{u} \cdot \vec{x})(\vec{v} \cdot \vec{w})\)
    4. Identities involving the gradient
    5. \(\vec{\nabla}(\phi+\Psi)=\vec{\nabla} \phi+\vec{\nabla} \psi\)
    6. \(\vec{\nabla}(\phi \psi)=\psi \vec{\nabla} \phi+\phi \vec{\nabla} \psi\)
    7. \(\vec{\nabla}(\vec{u} \cdot \vec{v})=[\vec{v} \cdot \vec{\nabla}] \vec{u}+\vec{v} \times(\vec{\nabla} \times \vec{u})+[\vec{u} \cdot \vec{\nabla}] \vec{v}+\vec{u} \times(\vec{\nabla} \times \vec{v})\)
    8. Identities involving the divergence
    9. \(\vec{\nabla} \cdot(\vec{u}+\vec{v})=\vec{\nabla} \cdot \vec{u}+\vec{\nabla} \cdot \vec{v}\)
    10. \(\vec{\nabla} \cdot(\phi \vec{u})=\vec{u} \cdot \vec{\nabla} \phi+\phi \vec{\nabla} \cdot \vec{u}\)
    11. \(\vec{\nabla} \cdot(\vec{u} \times \vec{v})=\vec{v} \cdot(\vec{\nabla} \times \vec{u})-\vec{u} \cdot(\vec{\nabla} \times \vec{v})\)
    12. Identities involving the curl
    13. \(\vec{\nabla} \times(\vec{u}+\vec{v})=\vec{\nabla} \times \vec{u}+\vec{\nabla} \times \vec{v}\)
    14. \(\vec{\nabla} \times(\phi \vec{u})=\vec{\nabla} \phi \times \vec{u}+\phi \vec{\nabla} \times \vec{u}\)
    15. \(\vec{\nabla} \times(\vec{u} \times \vec{v})=[\vec{v} \cdot \vec{\nabla}] \vec{u}-\vec{v}(\vec{\nabla} \cdot \vec{u})-[\vec{u} \cdot \vec{\nabla}] \vec{v}+\vec{u}(\vec{\nabla} \cdot \vec{v})\)
    16. \(\vec{\nabla} \times(\vec{\nabla} \times \vec{u})=\vec{\nabla}(\vec{\nabla} \cdot \vec{u})-\nabla^{2} \vec{u}\)
    17. \(\vec{\nabla} \cdot(\vec{\nabla} \times \vec{u})=0\)
    18. \(\vec{\nabla} \times(\vec{\nabla} \phi)=0\)
    19. Identities involving the Laplacian
    20. \(\nabla^{2}(\phi \psi)=\psi \nabla^{2} \phi+\phi \nabla^{2} \psi+2 \vec{\nabla} \phi \cdot \vec{\nabla} \psi\)
    21. \(\nabla^{2}(\phi \vec{u})=\vec{u} \nabla^{2} \phi+\phi \nabla^{2} \vec{u}+2(\vec{\nabla} \phi) \cdot \vec{\nabla} \vec{u}\)
    22. Identities involving the advective derivative
    23. \([\vec{u} \cdot \vec{\nabla}](\phi \vec{v})=(\vec{u} \cdot \vec{\nabla} \phi) \vec{v}+\phi([\vec{u} \cdot \vec{\nabla}] \vec{v})\)
    24. \([\vec{u} \cdot \vec{\nabla}](\vec{v} \cdot \vec{w})=([\vec{u} \cdot \vec{\nabla}] \vec{v}) \cdot \vec{w}+\vec{v} \cdot([\vec{u} \cdot \vec{\nabla}] \vec{w})\)
    25. \([\vec{u} \cdot \vec{\nabla}](\vec{v} \times \vec{w})=([\vec{u} \cdot \vec{\nabla}] \vec{v}) \times \vec{w}+\vec{v} \times([\vec{u} \cdot \vec{\nabla}] \vec{w})\)
    26. \([\vec{u} \cdot \vec{\nabla}] \vec{u} \equiv(\vec{\nabla} \times \vec{u}) \times \vec{u}+\frac{1}{2} \vec{\nabla}(\vec{u} \cdot \vec{u})\)

    16: Appendix E- Vector Identities is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Bill Smyth via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.