We have $\omega = (0,0, \xi(x,y,t))$ and $\textbf u =(u(x,y,t),v(x,y,t),0)$ and that $$\frac{\partial \xi}{\partial t} +u \frac{\partial \xi}{\partial x} +v\frac{\partial \xi}{\partial y}=0$$ is a simplified vorticity equation. Use $\nabla \cdot \textbf u =0$ to show $$\frac{\partial \xi^n}{\partial t} + \frac{\partial (u \xi^n)}{\partial x} +\frac{\partial (v \xi^n)}{\partial y}=0$$
Can someone show me a solution to this please because the one i have is that they multiply the simplified equation by $n \xi ^{n-1}$ and then the $n$ suddenly seems to disappear and then some product rule but it doesn't make sense to me.
$$H=\frac{\partial \xi^n}{\partial t} + \frac{\partial (u \xi^n)}{\partial x} +\frac{\partial (v \xi^n)}{\partial y}$$
Now use the product rule
$$H=\frac{\partial \xi^n}{\partial t} + \frac{\partial u}{\partial x}\xi^n+u\frac{\partial \xi^n}{\partial x} +\frac{\partial v}{\partial y}\xi^n+v\frac{\partial \xi^n}{\partial y}$$
Rearrange terms:
$$H=\left[\frac{\partial \xi^n}{\partial t}+u\frac{\partial \xi^n}{\partial x}+v\frac{\partial \xi^n}{\partial y}\right] +\left[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right]\xi^n$$
Using chain rule for $\xi^n$: $$H=n\xi^{n-1}\left[\frac{\partial \xi}{\partial t}+u\frac{\partial \xi}{\partial x}+v\frac{\partial \xi}{\partial y}\right] +\left[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right]\xi^n$$
In the last expression the first bracket is zero because of 1.PDE and the second bracket is zero because $\nabla \cdot u=0$. You can conclude $H=0$