Curl of a vector function. Hello friends, today it’s about the curl of a vector function.

Have a look!!

### Curl of a vector function

Suppose I have a vector such as .

Now the curl of this vector will be

So if I evaluate the determinant, I’ll get the curl of this 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 and so on. Also, pretty soon I’ll write about the divergence of any vector.

Now I’ll give some examples.

#### Examples of the curl 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 Stroud and Booth (2011)*, “Show that curl is a constant vector.”

##### Solution

Now here the given vector is .

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

So I can say . And this means .

Now, 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

So this gives

Now I’ll simplify it to get

And is obviously a constant vector.

Hence I can prove that the curl of the vector is a constant vector.

So this is the answer to this example.

Next, I’ll give another example.

##### Example 2

According to Stroud and Booth (2011)*, “If , find curl curl .”

##### Solution

Now in this example, the given vector is .

First of all, I’ll find out the curl of the vector .

###### Step 1

So the curl of the vector will be

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

So this gives

Now I’ll simplify it to get

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

###### Step 2

Thus curl curl will be will be

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

So this gives

Now I’ll simplify it to get

Hence I can conclude that curl curl 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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