Divergence of a vector function. Hello friends, today it’s about the divergence of a vector function.

Have a look!!

### Divergence of a vector function

Suppose I have a vector such as .

Now the divergence of this vector will be

So if I use the technique for first-order partial differentiation of functions with three variables, I will get the divergence of the vector.

If interested, you can read more about the other posts in vector analysis like directional derivative, the gradient of a scalar field, unit normal vector, unit tangent vector, curl of any vector and so on.

Now I’ll give some examples.

#### Some examples of the divergence of a vector function

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 Kreyszig (2005)*, “Find the divergence of the following vector function: .”

##### Solution

Now here the given vector is .

First of all, I’ll give it a name, say, .

So, in vector form, it will be

As per the the formula for the divergence of any vector, divergence of will be

Thus it will be

So this means

Now I’ll simplify it to get

which means

Hence I can conclude that this is the solution to the given example.

Now I’ll give another example.

##### Example 2

According to Kreyszig (2005)*, “Find the divergence of the following vector function: .”

##### Solution

Now here the given vector is .

First of all, give it a name, say, .

So, in vector form, it will be

Thus the vector is

As per the the formula for the divergence of any vector, divergence of will be

Thus it will be

So this means

Now I’ll simplify it to get

which means

Hence I can conclude that this is the answer to the given example.

Now comes my last example.

##### Example 3

According to Kreyszig (2005)*, “Find the divergence of the following vector function: .”

##### Solution

Now here the given vector is .

First of all, give it a name, say, .

Thus in vector form, it will be

So the divergence of will be

Therefore it will be

So this means

which gives

Hence I can conclude that this is the answer to the given 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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