# What algebraic manipulations make this two equations equivalent?

I have a very basic mathematical backround. I kind of understand the concept of a partial derivative but i dont know what algebraic manipulations make this two equations equivalent (the ones on 2A.4) given that density=constant/V.

• Which book ??? (on fluid mechanics ???) Commented Jan 12, 2023 at 20:49
• @Frobenius This is a truly amazing book "THE TRAGICOMICAL HISTORY OF THERMODYNAMICS" by Truesdell. I strongly recommend it both for its content and very entertaining style, but is surely not an "introduction" to the subject. Commented Jan 13, 2023 at 13:11
• @hyportnex : Many Thanks for the information. Commented Jan 13, 2023 at 14:44

To solve this problem you need to apply the chain rule to express the derivative $$\frac{\partial}{\partial \rho}$$ by $$\frac{\partial}{\partial V}$$. Using the relation between $$\rho$$ and $$V$$, we first calculate
$$\rho=M/V \Leftrightarrow V=M/\rho \quad \text{and} \quad \frac{\partial V}{\partial \rho}=-\frac{M}{\rho^2}=-\frac{V^2}{M}\,.$$
$$\rho \frac{\partial p(\rho,\theta)}{\partial \rho}=(M/V)\frac{\partial V}{\partial \rho}\frac{\partial p(V,\theta)}{\partial V}=(M/V)\left(-\frac{V^2}{M}\right)\frac{\partial p(V,\theta)}{\partial V}=-V \frac{\partial p(V,\theta)}{\partial V}\,.$$