# Why is a multivariable function satisfying the following conditions exact?

I read this section in the book "Mathematics Methods for Physics and Engineering"

Determining whether a differential containing many variable $$x_1,x_2,...,x_n$$ exact is a simple extension of the above. A differential containing many variables can be written in general as $$df=\Sigma^n_{i=1}g_i(x_1,x_2,...,x_n)dx_i$$ and the function will be exact if $$\frac{\partial g_i}{\partial x_j}=\frac{\partial g_j}{\partial x_i}$$ for all pairs of i and n.

I can understand the situation when there are only two variables in the function, but when there are many variables, I am a bit confused about why this condition is already necessary to prove that the function is exact, instead of requiring all partial derivatives of the function to be equal.

• By definition, an exact differential one form $$\omega=\sum^n_{i=1}g_i(x_1,x_2,...,x_n)\,dx_i$$ arises from a function, aka $$0$$-form, $$f$$ by exterior differentiation $$\omega=df\,.$$

• Saying that the differential $$\omega$$ is exact amounts to showing that the function $$f$$ exists. Writing the differential from the beginning as $$df$$ is therefore confusing.

• The reason for this confusion is imho that the authors Riley, Hobson & Bence are trying to avoid the language of differential forms and mention the above result without proof or reference.

• By the Poincare lemma it is enough to show that $$\omega$$ is closed which means that $$d\omega=0\,.$$

• By the exterior differential calculus rules, $$\begin{eqnarray} d\omega&=\sum^n_{i,j=1} \partial_j g_i(x_1,x_2,...,x_n)\,dx_j\wedge dx_i \end{eqnarray}$$ which is zero because $$\partial_j g_i$$ is symmetric in $$i,j$$ and $$dx_j\wedge dx_i$$ is antisymmetric.

• In order that the Poincare lemma is applicable we need to assume that the domain of the $$g_i$$ functions is simply connected.

• Examples of inexact differentials arise in Thermodynamics: change in heat $$Q$$ and change in work $$W$$ and are usually denoted by $$\delta Q$$ and $$\delta W$$ following Carl Gottfried Neumann. The first law of thermodynamics is then $$dU=\delta Q-\delta W$$ where $$U$$ is the internal energy and has an exact differential.

The author shouldn't write $$df$$ there. Suppose $$\Omega \subset \mathbb{R}^n$$ is an open set. A one form is an object of the form $$\omega = \sum_{i = 1}^{n}g_i(x)\,dx_i,$$ where each $$g_i \in C^{\infty}(\Omega, \mathbb{R})$$. The one-form $$\omega$$ is said to be exact if there exists a smooth function $$f \in C^{\infty}(\Omega, \mathbb{R})$$ such that $$df := \sum_{i = 1}^{n}\frac{\partial f}{\partial x_i}dx_i = \omega.$$ The above equation is equivalent to $$g_i = \frac{\partial f}{\partial x_i} \text{ for } i = 1, \dots, n.$$ Now if $$\omega$$ is exact, meaning that the above holds, then $$\frac{\partial g_i}{\partial x_j} = \frac{\partial^2 f}{\partial x_i \partial x_j} = \frac{\partial^2 f}{\partial x_j \partial x_i} = \frac{\partial g_j}{\partial x_i}.$$ So exactness implies $$\frac{\partial g_i}{\partial x_j} = \frac{\partial g_j}{\partial x_i}$$ for $$i, j \in \{1, \dots, n\}$$. However, the converse is not true in general. It is true when $$\Omega$$ is simply connected though, which means roughly that any smooth closed curve in $$\Omega$$ is smoothly homotopic to a point through a family of closed curves.