In this section we are going to introduce a couple of new concepts, the curl and the divergence of a vector.
Let’s start with the curl. Given the vector field the curl is defined to be,
There is another (potentially) easier definition of the curl of a vector field. We use it we will first need to define the operator. This is defined to be,
We use this as if it’s a function in the following manner.
So, whatever function is listed after the is substituted into the partial derivatives. Note as well that when we look at it in this light we simply get the gradient vector.
Using the we can define the curl as the following cross product,
We have a couple of nice facts that use the curl of a vector field.
Facts
1. If
2. If
3. If

Example 1 Determine if
Solution
So all that we need to do is compute the curl and see if we get the zero vector or not.
So, the curl isn’t the zero vector and so this vector field is not conservative.
