I am looking at Leveque's book on finite volume methods for hyperbolic problems. I understand the method, but for some reason I am having a little trouble understanding this particular algebraic manipulation below from page 17 of the book. Let's define $q(x, t)$ as the density of a chemical in a fluid, and $f(q)$ is a sufficiently smooth flux function. The idea is to derive the differential form of the PDE.
Maybe I am just forgetting my multivariate calculus, but I am trying to understand how the right hand side of this derivation leads to the subsequent statement.
So the function that is confusing me is $-\int_{x_1}^{x_2}\frac{\partial}{\partial x}f(q(x, t))dx $. How did the subsquent derivation get a $\frac{\partial}{\partial t}q(x, t)$ term, when the original partial derivative was only with respect to $x$. I would imagine that this term should be something like $\frac{d f}{d q} \frac{dq}{dx}$ The term from the book looks like a total differential, but I just wanted to confirm/understand how the total differential came about from the $-\int_{x_1}^{x_2}\frac{\partial}{\partial x}f(q(x, t))dx$ term.
It's an application of Leibniz integral rule on the left hand side term. Since q is smooth you can exchange the order of the integral and the derivative with respect with time. The term on the right hand side has not been affected.