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. To 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
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Example 1 Determine
if is a conservative vector field.
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.
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Next we should talk about a physical interpretation of the
curl. Suppose that is the velocity field of a flowing fluid. Then represents the tendency of particles at the
point to rotate about the axis that points in the
direction of . If then the fluid is called irrotational.
Let’s now talk about the second new concept in this
section. Given the vector field the divergence is defined to be,
There is also a definition of the divergence in terms of the
operator.
The divergence can be defined in terms of the following dot product.
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Example 2 Compute
for
Solution
There really isn’t much to do here other than compute the
divergence.
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We also have the following fact about the relationship
between the curl and the divergence.
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Example 3 Verify
the above fact for the vector field .
Solution
Let’s first compute the curl.
Now compute the divergence of this.
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We also have a physical interpretation of the
divergence. If we again think of as the velocity field of a flowing fluid then represents the net rate of change of the mass
of the fluid flowing from the point per unit volume. This can also be thought of as the tendency
of a fluid to diverge from a point. If then the is called incompressible.
The next topic that we want to briefly mention is the Laplace operator. Let’s first take a look at,
The Laplace operator is
then defined as,
The Laplace operator arises
naturally in many fields including heat transfer and fluid flow.
The final topic in this section is to give two vector forms
of Green’s Theorem. The first form uses
the curl of the vector field and is,
where is the standard unit vector in the
positive z direction.
The second form uses the divergence. In this case we also need the outward unit
normal to the curve C. If the curve is parameterized by
then the outward unit normal is given by,
Here is a sketch illustrating the outward unit normal for
some curve C at various points.
The vector form of Green’s Theorem that uses the divergence
is given by,