Line Integrals of Vector Fields

Definition

Suppose that a curve C is defined by the vector function r = r (s), 0 ≤ s ≤ S, where s is the arc length of the curve. Then the derivative of the vector function

is the unit vector of the tangent line to this curve (Figure 1).

Here and are the angles between the tangent line and the positive axis and respectively.

We introduce the vector function defined over the curve so that for the scalar function

the line integral exists. Such an integral is called the line integral of the vector field along the curve and is denoted as

Thus, by definition,

where is the unit vector of the tangent line to the curve

The latter formula can be written in the vector form:

where

If a curve lies in the -plane and , we can write:

Properties of Line Integrals of Vector Fields

The line integral of vector function has the following properties:

1. Let denote the curve which is traversed from to and let denote the curve with the opposite orientation − from to Then
   $$
2. If $$ is the union of the curves $$ and $$ (Figure $$), then
   $$



3. If the curve $$ is parameterized by $$ $$ then
   $$
4. If $$ lies in the $$-plane and is given by the equation $$ (in this case $$ and $$), then the latter formula can be written as
   $$

Solved Problems

Solution.

Using the formula

we find the answer:

Solution.

To find the given integral, we use the formula

Substituting $$ and $$ in the integrand, we obtain