Del Dot Expansion

Question

I have an expansion question using the $\nabla$. If I have this equation: $\nabla \dot\ (V \dot \ \nabla V)$. Where $V = x$ and $y$ components of velocity.

How does this expand to $(\frac{du}{dx})^2 + 2\Big(\frac{du}{dy} \frac{dv}{dx}\Big) + \Big(\frac{dv}{dy}\Big)^2$?

Thanks for any guidance.

Answer 1

$\nabla.(V.\nabla V) = \nabla((x,y).(x',y')) = \nabla(xx'+yy')$ grad of this will be all this differentiated again ($\nabla f$ is just gradient of f)