# The derivation of the advection-diffusion equation uses $\nabla\cdot(c\vec{v})=(\vec{v}\cdot\nabla)c$. Why doesn't the order of the derivative matter?

In a derivation of the advection-diffusion equation, it is exploited that $$\vec{\nabla} \cdot (c \vec{v}) = ( \vec{v}\cdot \vec{\nabla})c$$, where $$\vec{v}$$ and c respectively are the velocity and the concentration. How can the order of the gradient not matter?

• Work it out -- use the chain rule to expand the left hand side. What terms do you get? What assumptions are in your equations that allow you to keep or eliminate terms? – tpg2114 Dec 17 '18 at 17:52

If $$c$$ and $$\vec v$$ is an arbitrary pair of functions, then the identity you wrote is false; instead it must read $$\nabla \cdot (c\vec v) = (\vec v\cdot \nabla) c + c (\nabla \cdot \vec v),$$ which is easy to prove component-wise.
If your text is disregarding the second term, then presumably they're working under conditions where $$\nabla \cdot \vec v = 0$$. That's a natural assumption to have if $$\vec v$$ is the velocity field of an incompressible static flow, but you'll need to check your text for exactly what reasoning underlies that assumption.