If I have the expression as shown below:
$$ dq - p_{ext} dV = ( \frac{\partial U}{\partial V})_{T} dV + ( \frac{\partial U}{\partial T})_{V} dT$$
Can I say that:
$$ (\frac{ \partial U}{\partial V})_T = -P_{ext}$$
...by comparing differentials?
This is a commonly used relation in many thermodynamics textbook but I'm not sure how to mathematically justify this procedure.
For those who are not familiar with thermo, there is usually a local implicit relation between the temperature, volume and pressure quantites.