Fundamental theorem of line integrals
The fundamental theorem of calculus gives a relation between the integral of the derivative of a function and the value of the function at the boundary, that is,
The aim is now to find an analogous formula for line integrals.
We have seen previously in the section on vector line integrals that the line integral of a vector field over a curve is given by
The last integral is used for evaluating line integrals and is of the form (1). Thus, if there is a function such that
then we can use (1) directly to obtain a fundamental theorem of line integrals. Hence the question arises for which vector fields and curves this property holds.
Notice that Equation (1) holds for all, say, continuously differentiable functions and all intervals (where is continuously differentiable). Similarly, we want property (2) to hold for all curves and only put a restriction on the vector field (to be more precise: all curves whose range is in a region where the property of holds).
Fundamental Theorem of Line Integrals
Let be a continuously differentiable function and let be a continuously differentiable parameterisation of a curve . Then
Note that the same result holds for and curves in which satisfy the assumptions.
Let be a continuously differentiable function.
- Let be a closed curve. Then
- Let and be two curves with the same starting and end points. Then
Let Calculate along a curve with starting point and end point .
Notice that the fundamental theorem of calculus only holds for vector fields which are gradient fields, i.e. for which there exists a function such that .
The following example shows that the fundamental theorem of calculus does not hold in if the vector field is not defined everywhere.
Let be given by . Let the curve be the unit circle around the origin, which is parameterised by given by . Calculate .
A vector field for which there exists a function such that is called conservative. The function is called potential function.
The gravitational force acting on a mass due to a mass is given by
where is the gravitational constant, is a unit vector pointing from to and is the distance between the two mass points.
Two questions now arise:
- Given a vector field . How can we find out whether is conservative?
- Assuming we know that is conservative, how can we find a function such that
To answer the first question, recall that in Section 2 on divergence and curl that Let be twice continuously differentiable. Then, if there is a function such that , then
Thus, we can test whether a given vector field is conservative by calculating its curl. If then we know that is not conservative.
Show that the vector field is not conservative.
The converse also holds under some assumptions. This is a special case of Poincare’s lemma. For instance, Poincare’s lemma implies that if is twice continuously differentiable and everywhere, then is conservative.
Let . Find a function such that .
Let Find a function such that .