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In the previous two sections we looked at line integrals of
functions. In this section we are going
to evaluate line integrals of vector fields.
We’ll start with the vector field,
and the three-dimensional, smooth curve given by
|
The line integral
of  along C
is
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Note the notation in the left side. That really is a dot product of the vector field and the
differential and the differential really is a vector. Also, is a shorthand for,
We can also write line integrals of vector fields as a line
integral with respect to arc length as follows,
where is the unit tangent vector and is given by,
If we use our knowledge on how to compute line integrals
with respect to arc length we can see that this second form is equivalent to
the first form given above.
In general we use the first form to compute these line
integrals as it is usually much easier to use.
Let’s take a look at a couple of examples.
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Example 1 Evaluate
where and C
is the curve given by ,
.
Solution
Okay, we first need the vector field evaluated along the
curve.
Next we need the derivative of the parameterization.
Finally, let’s get the dot product taken care of.
The line integral is then,
|
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Example 2 Evaluate
where and C
is the line segment from and .
Solution
We’ll first need the parameterization of the line
segment. We saw how to get the
parameterization of line segments in the first section
on line integrals. We’ve been using
the two dimensional version of this over the last couple of sections. Here is the parameterization for the line.
So, let’s get the vector field evaluated along the curve.
Now we need the derivative of the parameterization.
The dot product is then,
The line integral becomes,
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Let’s close this section out by doing one of these in
general to get a nice relationship between line integrals of vector fields and
line integrals with respect to x, y, and z.
Given the vector field and the curve C parameterized by ,
the line integral is,
So, we see that,
Note that this gives us another method for evaluating line
integrals of vector fields.
This also allows us to say the following about reversing the
direction of the path with line integrals of vector fields.
Fact
This should make some sense given that we know that this is
true for line integrals with respect to x,
y, and/or z and that line integrals of vector fields can be defined in terms
of line integrals with respect to x, y, and z.