How to calculate the variation of the function $\Phi$ below, whose arguments are a vector and a scalar field, which occurs sometimes in fluid dynamics?
$$ \Phi \left ( \vec{u}\,,\rho \right) = \bigtriangledown \times \vec{u}\cdot \bigtriangledown \rho\label{1}\tag{1} $$ where
- $\vec{u}$ is a vector field as flow velocity in flow dynamics,
- $\rho$ is a scalar field as density,
- $\bigtriangledown$ is gradient operator.
As said in the introduction to the questions, $\Phi ( \vec{u}\,, \rho)$ is a function of both $\vec{u}$ and $\rho$, thus the couple $(\vec{u}, \rho)$ should be "varied" in order to calculate its variation.
In sum, my question is how to calculate the variation of \eqref{1} i.e.
$$ \delta \Phi ( \vec{u}\,, \rho) = A(\vec{u}\,, \rho)\delta\vec{u}\:+B(\vec{u}\,, \rho)\delta\rho+C(\vec{u}\,, \rho) $$ $$ A(\vec{u}\,, \rho)=? $$ $$ B(\vec{u}\,, \rho)=? $$ $$ C(\vec{u}\,, \rho)=? $$