MTH2010 Chap.10 Conservative Fields and Green's Theorem
Conservative Fields and Green's Theorem
Week 10 lists four topics: conservative vector fields, the fundamental theorem for line integrals, path independence, and Green's theorem. A vector field is conservative when it is the gradient of a scalar function, called a potential.
For such a field the fundamental theorem for line integrals replaces a line integral by two evaluations of the potential: the integral of the gradient along any curve from A to B equals the potential at B minus the potential at A. Line integrals of a conservative field therefore depend only on the endpoints, and around any closed curve they are zero.
In the plane, a field (P, Q) with continuous partial derivatives on a region without holes is conservative exactly when P_y equals Q_x, and a potential can be built by integrating one component and matching the other. The hypothesis about holes is essential, and the standard example of a field that passes the component test away from the origin but still circulates around it shows why.
Green's theorem connects a line integral around the boundary of a plane region with a double integral over the region: the integral of P dx + Q dy around the positively oriented boundary equals the integral of Q_x - P_y over the region. Positive orientation keeps the region on the left, which for a region with a hole means the inner boundary is traversed clockwise.
Choosing P and Q suitably also turns Green's theorem into a formula for area.
What this chapter covers
- 01
Conservative fields and potentials
- 02
Fundamental theorem: potential at the end minus potential at the start
- 03
Path independence and zero circulation around closed curves
- 04
Component test P_y = Q_x and the no-holes hypothesis
- 05
Building a potential by integrating and matching
- 06
Green's theorem: boundary circulation equals an area integral
- 07
Positive orientation, including inner boundaries
- 08
Area from a boundary integral
Worked example · free
Green's theorem on a triangle
- 1Here P = y and Q = 3x, so Q_x - P_y = 3 - 1 = 2.
- 1The triangle is positively oriented and has area (1/2)(2)(2) = 2.
- 1Green's theorem: the line integral equals the integral of 2 over the triangle, which is 2 times 2 = 4.
Key terms
- Potential function
- A scalar function whose gradient equals a given vector field.
- Path independence
- The property that a line integral depends only on the endpoints of the curve, not on the route.
- Positive orientation
- The direction of travel along a boundary that keeps the enclosed region on the left.
- Simply connected region
- A connected region in the plane with no holes, on which the component test guarantees a potential.
Conservative Fields and Green's Theorem FAQ
How do I test whether a plane vector field is conservative?
Check that P_y equals Q_x and that the field's domain has no holes. If both hold, a potential exists; build it by integrating P in x and matching the y-derivative with Q.
Why must the domain have no holes?
A field can satisfy P_y = Q_x everywhere except at a point and still circulate around that point, giving a non-zero integral around a loop. The hole breaks the link between the test and conservativeness.
Which way should a boundary be traversed in Green's theorem?
Positively, so the region stays on your left. For a disc that means anticlockwise; for a region between two curves the outer boundary goes anticlockwise and the inner one clockwise.
How can Green's theorem give an area?
Choose P and Q with Q_x - P_y = 1, for example P = -y/2 and Q = x/2. The double integral is then the area, and the boundary integral computes it.
Exam move
Learn the three equivalent descriptions of a conservative field together: gradient of a potential, path independence, and zero circulation on closed curves. For every line integral, test for a potential before parametrising, and for every closed plane curve, consider Green's theorem first. Practise building potentials until the matching step is reliable, then check each potential by differentiating it.
For Green's theorem, draw the orientation arrows on a sketch before computing, especially for regions with holes.
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