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16: Appendix E- Vector Identities

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

    This page titled 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 the style and standards of the LibreTexts platform.