16: Appendix E- Vector Identities


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. $$(\vec{u} \times \vec{v})(\vec{w} \times \vec{x})=(\vec{u} \cdot \vec{w})(\vec{v} \cdot \vec{x})-(\vec{u} \cdot \vec{x})(\vec{v} \cdot \vec{w})$$
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. $$[\vec{u} \cdot \vec{\nabla}] \vec{u} \equiv(\vec{\nabla} \times \vec{u}) \times \vec{u}+\frac{1}{2} \vec{\nabla}(\vec{u} \cdot \vec{u})$$

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; a detailed edit history is available upon request.