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

### Determine the exact differential in two variables

If any function is dependent on two real variables and , then I’ll write the function as .

Now the differential of the function is where .

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

Therefore the differential will be an exact differential if

But if the function depends on three variables, say, and , then the conditions will be different. And that I’ll discuss in some other post. Today the focus is on two variables.

Ok, now I’ll give some examples.

#### Some examples of how to determine exact differential in two variables

Note: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example.

##### Example 1

According to Stroud and Booth (2011)* “Determine which is the exact differential: .”

##### Solution

So here the given differential is

Now I compare it with the standard form of the differential So I can say that

First of all, I’ll differentiate partially with respect to . So this gives

(1)

Next, I’ll differentiate partially with respect to . So this gives

(2)

As I can see from both equations (1) and (2) that

Hence I can conclude that the differential is an exact differential. And this is the answer to this example.

Now I’ll give another example.

So here is my second example.

##### Example 2

According to Stroud and Booth (2011)* “Determine whether is an exact differential.”

##### Solution

Now here the given differential is

Again I’ll compare it with the standard form of the differential So I can say that

First of all, I’ll differentiate partially with respect to . So this gives

(3)

Next, I’ll differentiate partially with respect to . So this gives

(4)

As I can see from both equations (3) and (4) that

Hence I can conclude that the differential is an exact differential. And this is the answer to this example.

Now I’ll give another example.

So here is my last example.

##### Example 3

According to Stroud and Booth (2011)* “Verify that is an exact differential.”

##### Solution

Now here the given differential is

Again I’ll compare it with the standard form of the differential So I can say that

First of all, I’ll differentiate partially with respect to . So this gives

(5)

Next, I’ll differentiate partially with respect to . So this gives

(6)

As I can see from both equations (5) and (6) that

Hence I can conclude that the differential 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!!

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