How to get the exact differential in three variables %

How to get the exact differential in three variables. Hello friends, today I’ll show how to determine the exact differential in three variables. Have a look!!

How to get the exact differential in three variables

If any function $w$ is dependent on three real variables $x, y$ and $z$ , then I’ll write the function as $w = f(x, y, z)$ .

Now the differential $\text{d}w$ of the function $w$ is $\text{d}w = \text{P}~\text{d}x+\text{Q}~\text{d}y+\text{R}~\text{d}z$ where $\text{P} = \cfrac{\partial w}{\partial x}, \text{Q} = \cfrac{\partial w}{\partial y}, \text{R} = \cfrac{\partial w}{\partial z}$ .

If interested, you can read more on how to get the differential dz of a function.

Therefore the differential $\text{d}z$ will be an exact differential if

$\boxed{\cfrac{\partial \text{P}}{\partial y} = \cfrac{\partial \text{Q}}{\partial x},~~~ \frac{\partial \text{P}}{\partial z} = \cfrac{\partial \text{R}}{\partial x},~~~ \cfrac{\partial \text{R}}{\partial y} = \cfrac{\partial \text{Q}}{\partial z}.}$

If interested, you can also read more about the exact differential in two variables.

Now I’ll give an example of how to determine the exact differential in three variables.

An example of how to determine exact differential in three variables

Note: This example is not mine. I have chosen it from some book. I have also given the due reference at the end of the post.

So here is my example.

Example

According to Stroud and Booth (2011)* “Verify that $\text{d}w = \cfrac{y}{z}~\text{d}x+\cfrac{x}{z}~\text{d}y-\cfrac{xy}{z^2}~\text{d}z$ is an exact differential…”

Solution

So here the given differential is

$\boxed{\text{d}w = \frac{y}{z}\text{d}x+\frac{x}{z}\text{d}y-\frac{xy}{z^2}\text{d}z.}$

Now I compare it with the standard form of the differential $\text{d}w = \text{P}~\text{d}x+\text{Q}~\text{d}y+\text{R}~\text{d}z.$ So I can say that

$\boxed{\text{P} = \frac{y}{z},~~~ \text{Q} = \frac{x}{z},~~~ \text{R} = -\frac{xy}{z^2}.}$

First of all, I’ll differentiate $\text{P}$ partially with respect to $y$ . So this gives

(1) $\begin{equation*}\frac{\partial \text{P}}{\partial y} = \frac{1}{z}.\end{equation*}$

Next, I’ll differentiate $\text{P}$ partially with respect to $z$ . So this gives

(2) $\begin{equation*}\frac{\partial \text{P}}{\partial z} = -\frac{y}{z^2}.\end{equation*}$

Then I’ll differentiate $\text{Q}$ partially with respect to $x$ . And this gives

(3) $\begin{equation*}\frac{\partial \text{Q}}{\partial x} = \frac{1}{z}.\end{equation*}$

Now I’ll differentiate $\text{Q}$ partially with respect to $z$ . And this gives

(4) $\begin{equation*}\frac{\partial \text{Q}}{\partial z} = -\frac{x}{z^2}.\end{equation*}$

Next, I’ll differentiate $\text{R}$ partially with respect to $x$ . And this gives

(5) $\begin{equation*}\frac{\partial \text{R}}{\partial x} = -\frac{y}{z^2}.\end{equation*}$

Finally, I’ll differentiate $\text{R}$ partially with respect to $y$ . So this means

(6) $\begin{equation*}\frac{\partial \text{R}}{\partial y} = -\frac{x}{z^2}.\end{equation*}$

As I can see from both equations (1) and (3) that

$\boxed{\frac{\partial \text{P}}{\partial y} =\frac{\partial \text{Q}}{\partial x} =\frac{1}{z}.}$

Also I can compare both equations (2) and (5) to see that

$\boxed{\frac{\partial \text{P}}{\partial z} =\frac{\partial \text{R}}{\partial x} =-\frac{y}{z^2}.}$

Again I can see from both equations (4) and (6) that

$\boxed{\frac{\partial \text{Q}}{\partial z} =\frac{\partial \text{R}}{\partial y} =-\frac{x}{z^2}.}$

Hence I can conclude that the differential $\text{d}w = \cfrac{y}{z}~\text{d}x+\cfrac{x}{z}~\text{d}y-\cfrac{xy}{z^2}~\text{d}z$ is an exact differential. And this is the answer to this example.

Dear friends, this is the end of my today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

*Reference: K. A. Stroud and Dexter J. Booth (2011): Advanced engineering mathematics, Industrial Press, Inc.; 5th Edition (March 8, 2011), Chapter: Multiple integration 1, Further problems 19, p.690, Q. No. 11.

How to get the exact differential in three variables