Combinations of curl, grad, div in the vector and scalar fields. Hello friends, today I’ll show how to use the combinations of curl, grad, and div in the vector and scalar fields. Have a look!!

If you’re looking separately for curl, grad, div in the vector and scalar fields, do check out:

**The gradient vector (grad) of a scalar field**

**How to get the divergence vector (div) of a vector field**

**Combinations of curl, grad, div in the vector and scalar fields**

Now I’ll give you some examples.

**Solved examples**** of the curl, grad, div in the vector and scalar fields**

Disclaimer: 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 example.

**Example **

According to Stroud and Booth (2011)*, “If ; and ; determine, at the point

”

**Solution**

Now here the two vectors are and . Also the scalar field is . And my tasks are to get the values of and .

So I’ll start with .

**(d)**

First of all, I’ll get , that is, the divergence of the vector . Then I’ll find out the gradient vector of the .

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

So this means

Now I’ll get the gradient vector of the .

Thus it will be

Since , this means

Therefore at the point , the value of will be

So this means

Hence I can conclude that this is the answer to this part.

Now I’ll do the other part (e).

**(e)**

Next, I’ll find out the value of . So I’ll start with the value of .

According to the formula for the curl of a vector, curl of the vector will be

Next, I’ll evaluate this determinant to get the curl as

Now I’ll simplify it. So I can say that

Then I’ll get the curl of curl , that is, curl of the vector .

Thus curl curl will be

Next, I’ll evaluate this determinant to get the curl curl . So this gives

Now I’ll simplify it. Thus I get

Therefore at the point , the value of will be

Next, I’ll simplify it. So that means

Hence I can conclude that this is the answer to the part (e) of the given example.

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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