I am reading a paper about pde but I am stuck at the following expression:
Let $ X^t(\alpha):\mathbb R^d\rightarrow \mathbb R^d $ and $ t\in \mathbb R $
Assume that $$ \frac{d}{dt}X^t(\alpha)=v(X^t(\alpha),t), \quad X^0(\alpha)=\alpha $$
Let $ Q(\alpha,t)=det(\nabla_{\alpha}X^t(\alpha))$ ,then Q satisfies $$ \frac{d}{dt}Q(\alpha,t)= ( \text{div} v)(X^t(\alpha),t)Q(\alpha,t) $$
I cannot understand why the derivitive of Q with respect to t can be expressed like this.
Can anyone gives me some hints to help me understand this