# Confusion regarding: exact differentials, inexact differentials, directional derivative and partial derivatives. What is the proper approach?

When studying multivariable calculus we got introduced to partial derivatives and concepts like directional derivative. This all made sense. The notation was also quite clear to understand e.g.:

$$f(x,y) = x^2 + y^2$$ $$\frac{\partial f}{\partial x} = 2x$$ It's actually quite similar to one variable calculus, you just have to take the other variables as constant.

Intuitively it also makes sense, it is the slope in the $$x,y,z$$-direction. The gradient operator is also clear.

$$\nabla f = f_x \hat x + f_y \hat y + f_z \hat z$$

Where the directional derivative is used to specify a path along a certain vector:

$$\nabla_{\vec{v}} f = \nabla f \cdot \vec{v}$$

However I am studying thermodynamics at the moment and there is a lot of partial derivatives with none of the usual notation we used in multivariable calculus. E.g.:

$$\frac{\partial P}{\partial V} |_T$$ I.e. the partial deritative of pressure to volume while keeping temperature constant.

So suddenly it is important to remind the reader which variable is taken constant. I don't really understand why this is different than the usual multivariable notation $$f_x$$ where it is not common to do. It probably has to do with it being dependent/independent variables, but I don't see clearly why the notation and approach is different. I also don't really understand what it means to be independent and dependent. Is there an approach where you can just use the usual multivariable notation with perhaps directional derivatives? Is there something I am missing?

I strongly have the feeling that the confusion is because of inconsistent/different approach to the same problem. I am quite confused, I hope someone can reduce the confusion

• It has to do with thermodynamics being able to describe their functions equivalently through various different combinations of independent variables. I do agree that a lot of thermodynamics literature is probably a bit less rigorous than, say, mechanics or general relativity. But there are now even books that handle the subject using differential forms. Commented May 3, 2022 at 11:55
• Since the bar is usually used to signify at what point you evaluate the derivative, e.g., $\frac{\partial P}{\partial V}\big|_{V=V_0, T=T_0}$, the preferred notation, which I've seen without exception, is to use parentheses with the subscript: $$\left(\frac{\partial P}{\partial V}\right)_T$$ denotes the derivative with $T$ held constant. See, for example, this post. Commented May 3, 2022 at 17:10

It is only a peculiar notation that is unusual in mathematics. For example, $$\frac{\partial P}{\partial V} |_T$$ is the partial derivative of the function $$P=P(T,V)$$ with respect to $$V$$. The other variables, in this case only$$T$$, remain constant. If you want, this has also the purpose of reminding the reader that the only other variable is $$T$$. But of course, we should simply write $$\frac{\partial P}{\partial V}$$ or, in a bit more pedantic but mathematical manner, $$\frac{\partial P}{\partial V} (T,V).$$