Originally Posted by **Donny** Hi mscfd,
Thank you for your suggestion. I read the book and I can see the point. However, I am lost in translation in equation (45.1). I don't know how to elaborate delta f become delta x and delta y. |

Donny, eq. 45.1 is the definition of the total derivative. You can easily find a topic about this content in a Calculus book.

$\displaystyle f(x_1,x_2,x_2,....,x_n)$

$\displaystyle df=dx_1\frac{\partial f}{\partial x_1}+dx_2\frac{\partial f}{\partial x_2}+dx_3\frac{\partial f}{\partial x_3}+.....+dx_n\frac{\partial f}{\partial x_n}$

Anyway, if the internal energy can be defined as a function of the temperature and volume (this is not the only form): $\displaystyle U(T,V)$

$\displaystyle dU=dT\left(\frac{\partial U}{\partial T}\right)_V+dV\left(\frac{\partial U}{\partial V}\right)_T$

The letters under the parentheses mean that the variation of the derivative is for V or T constant. In pure mathematics this information is redundant, since it is a partial derivative, for a physicist or an engineer who is in the laboratory measuring, it is important to know which property should remain constant during the variation.