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

**Determine the exact differential**

Let’s suppose any function is dependent on two real variables and . Then I’ll write the function as .

Now the differential of the function is . And the values of and are and respectively.

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.

So let’s suppose the function is dependent on three real variables and . Then I’ll write the function as .

Thus the differential of the function will be . And the values of and will be .

Therefore the differential will be an exact differential if

Now I’ll give several examples on how to determine the exact differential in two and three variables.

#### Ex**amples **

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

Now I’ll simplify it. So this gives

(1)

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

(2)

Now I compare both equations (1) and (2). And that gives me

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

Now I’ll give you another 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 . And this gives

Then I’ll simplify it. So it becomes

(4)

Now I compare both equations (3) and (4). And that gives me

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

Next, comes my other 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 do it in the same way as in Examples 1 and 2. So I’ll compare it with the standard form of the differential Therefore I can say that

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

Now I’ll simplify it. And that gives

(5)

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

Then I’ll simplify it. Therefore it becomes

(6)

Now I compare both equations (5) and (6). And that gives me

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

Now I’ll show how to determine the exact differential in three variables.

**Example **

According to Stroud and Booth (2011)*1 “Verify that is an 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

(7)

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

(8)

Then I’ll differentiate partially with respect to . And this gives

(9)

Now I’ll differentiate partially with respect to . And this gives

(10)

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

(11)

Finally, I’ll differentiate partially with respect to . So this means

(12)

As I can see from both equations (7) and (9) that

Also I can compare both equations (8) and (11) to see that

Again I can see from both equations (10) and (12) that

Dear friends, this is the end of 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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