Exact and inexact differentials

An exact differential is a differential equation $du$ , e.g. of two variables, of the form $du=M(x,y)dx+N(x,y)dy$ , where $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ . For example, the total differential of $u=xy+c$ , where $c$ is a constant, is $du=\left(\frac{\partial u}{\partial x}\right)_ydx+\left(\frac{\partial u}{\partial y}\right)_xdy$ . Comparing this with the general form, we have $M(x,y)=\left(\frac{\partial u}{\partial x}\right)_y$ and $N(x,y)=\left(\frac{\partial u}{\partial y}\right)_x$ . Since $\frac{\partial M}{\partial y}=\frac{\partial^2u}{\partial y\partial x}=1=\frac{\partial^2u}{\partial x\partial y}=\frac{\partial N}{\partial x}$ , $du$ is an exact differential. The equality of the mixed partials $\frac{\partial^2u}{\partial y\partial x}=\frac{\partial^2u}{\partial x\partial y}$ implies that the change in $u$ is independent of the path taken.

Conversely, an inexact differential is a differential equation $du$ of the form $du=M(x,y)dx+N(x,y)dy$ , where $\frac{\partial M}{\partial y}\neq\frac{\partial N}{\partial x}$ . The change in $u$ , in this case, is dependent on the path taken.

Question

Show that $du=3ydx+xdy$ is an inexact differential.

Answer

We have $\frac{\partial M}{\partial y}=3$ and $\frac{\partial N}{\partial x}=1$ . So, $\frac{\partial M}{\partial y}\neq\frac{\partial N}{\partial x}$ .

Another difference between an exact differential and an inexact differential is that an exact differential integrates directly to give the function , whereas an inexact differential does not.

Question

Using the integral criterion, show that $du=xdx+ydy$ is an exact differential, while $du=3ydx+xdy$ is an inexact differential.

Answer

For the first differential, we have $\int du=u=xy+c$ . For the second differential, if we integrate with respect to $x$ , we have

$\int\frac{\partial u}{\partial x}dx=3\int ydx+\int x\frac{\partial y}{\partial x}dx$

Since $x$ and $y$ are independent, $\frac{\partial y}{\partial x}=0$ and we have $\int\frac{\partial u}{\partial x}dx=3\int ydx=3xy+c$ , where $c$ is only a constant with respect to $x$ . Therefore, we can rewrite $\int\frac{\partial u}{\partial x}dx=3xy+c$ as

$\int\frac{\partial u}{\partial x}dx=3xy+g(y)$

If we integrate the differential with respect to $y$ , we have

$\int\frac{\partial u}{\partial y}dy=3\int y\frac{\partial x}{\partial y}dy+\int xdy=xy+h(x)$

Clearly, $3xy+g(y)\neq xy+h(x)$ , for all $g(y)$ and $h(x)$ . Therefore, the second differential does not integrate directly to give a function.

The fact that an exact differential integrates directly to give the function $u$ but an inexact differential does not, implies that $M(x,y)$ and $N(x,y)$ for a differentiable function must be of the appropriate forms of $\left(\frac{\partial u}{\partial x}\right)_y$ and $\left(\frac{\partial u}{\partial y}\right)_x$ respectively. In other words, the total differential of a differentiable function must be an exact differential.