You are given the following velocity field for a flow $$\vec{v}(x,y, z, t) = (2axy) \vec{i} + a(b^2+x^2-y^2)\vec{j} + 0 \vec{k} $$where $a$ and $b$ are constants. Find $\nabla (v^2)$.
The issue I am having here is how on earth do I define $v^2$ from its vector field? I haven't ever come across a uniform definition for converting between a vector field and a scalar field.
In case it is relevant, this is in the context of the Navier-Stokes equations and the overall goal of the problem is to find an expression for $\nabla P$ (grad of the pressure). The solutions use $v^2 = (2axy)^2 + (a(b^2+x^2-y^2))^2$ and that is, of course, the obvious candidate, but I don't know if this is a convention?
[She also uses the identity: $$(\vec{v} \cdot \nabla) \vec{v} = \frac{1}{2} \nabla (v^2) -\vec{v} \times (\nabla \times \vec{v})$$ if that is helpful as well.]