Previously we considered integrals of a function (n=2,3) over a curve . We now turn to integrating vector fields where or . These integrals can be motivated by calculating the work done by a force on a particle along some curve .
In the simplest case, the work done a force acting on an object which moves along a straight line is given by
where the object is displaced in the direction of the vector with distance .
In a more general setting, the object moves along a curve . One way to arrive at a formula for calculating the work done by a force along is to take the component of in the tangential direction on each point on the curve and integrate this quantity using a scalar line integral. The tangential direction on the curve at a point is given by and a unit vector in tangential direction is given by
where denotes the Euclidian norm and where we assume that for all Hence the component of in the direction of the tangent to the curve at a point is given by
This is now a scalar valued function which can be integrated using the scalar line integral. We thus obtain that the work done is given by
where we used .
We can now formally define line integrals of vector fields over some curve.
Let or . Let be a continuously differentiable curve and let be a continuous vector field, where we assume that Then we define , the line integral of along , by the formula
Notice that we do not need the assumption that for all . This can be shown by setting up Riemann sums as in the case for scalar line integrals. (This makes a difference in some cases. For example, for we get , and this cannot be avoided using a different parameterisation.)
Line integrals are written in various forms. For instance, let and . Then the line integral is also written as
There also exist analogous ways for writing line integrals for the two-dimensional case.
Let for and . Then from the parameterisation of the curve we have and Hence and Further Hence
Some properties of line integrals
Let be continuous vector fields and let be a constant.
- Let be a smooth curve and let denote the same curve but with the orientation reversed. Then
- Let be a union of smooth curves , then
Line integrals in the plane in normal form
We defined line integrals by integrating the tangential component of a vector field over a curve . In the plane it is also meaningful to compute the orthogonal component of the vector field along some curve . Let with be a curve and let be a unit normal vector to the curve at the point which is obtained by turning the unit tangent vector by clockwise, then this line integral is given by