$\nabla({\bf u}^2)=2({\bf u}\cdot \nabla){\bf u} - 2(\nabla \times {\bf u}) \times {\bf u}$

Question

Please see the next link: http://www3.kis.uni-freiburg.de/~peter/teach/hydro/hydro02.pdf

In (2.13), he used: $$\nabla({\bf u}^2)=2({\bf u}\cdot \nabla){\bf u} - 2(\nabla \times {\bf u}) \times {\bf u}.$$

Is this is a formula? can someone let me some details/link about this formula?

Any help appreciated!

Answer 1

This is more like a maths question to me. This is just an identity, which is true and facilitates the calculation and it is valid for any vector field.

The proof, using Einstein summation convention would be something like:

$$ (\nabla \times \vec u )\times \vec u = \epsilon_{ijk}(\nabla \times u)_j u_k = \\ \epsilon_{ijk}\epsilon_{jlm}\partial_l (u_m) u_k = \epsilon_{jki}\epsilon_{jlm}\partial_l (u_m) u_k = \\ (\delta_{kl}\delta_{im} - \delta_{km}\delta_{il}) \partial_l (u_m) u_k = \\ \partial_k (u_i) u_k - \partial_i (u_k) u_k = (\vec u \cdot \vec \nabla) \vec u - \frac 1 2 \nabla(\vec u \cdot \vec u) $$

I hope that helps.