In Section 3 we established the Fundamental theorem of line integrals which gives a relationship between line integrals of conservative vector fields and the values of the potential function at the endpoints of the curve. Green’s theorem on the other hand gives a relationship between double integrals and line integrals in
Green’s theorem in normal form
Consider a simple closed curve in , that is we assume that the starting point coincides with the end point of the curve and that the curve does not intersect itself (see also the Jordan curve theorem). Let be continuously differentiable in the region whose boundary is . If we interpret as the velocity field of a fluid, then from the interpretation of line integrals in normal form we saw that can be understood as measuring the amount of fluid leaving the region . On the other hand, divergence can be interpreted as being the amount of fluid leaving a small square around a point per unit area. Hence the double integral over the region of the divergence is the amount of fluid leaving the area . In this interpretation, Green’s theorem states that
In mathematical terms, this is stated in the following theorem.
Green’s theorem in normal form
Let be a simple closed curve which can be parameterised by a piecewise continuously differentiable function and let be the region in enclosed by . Let be continuously differentiable. Then
where is the normal vector pointing outwards.
If . Assume that is oriented counterclockwise (or oriented positively). Then the tangent vector to is given by and the normal vector pointing outwards is therefore , which is obtained by turning by clockwise. Then
Hence the above formula can also be written as
Green’s theorem in tangential form
In this version of Green’s theorem the line integral is over the tangential component of the vector field rather then the normal component. It can be obtained by a simple substitution. Let be a continuously differentiable vector field and assume that the simple closed curve is oriented counterclockwise. The aim is to find a relationship between the line integral
and some double integral.
We can obtain such a formula from (1) by using the substitution and . That is, given the vector field , we define a vector field which is obtained by turning by clockwise. Then the line integral in tangent form over is the same as the line integral in normal form with pointing outwards over (since is oriented counterclockwise), that is
Green’s theorem in tangent form
Let be a simple closed curve oriented positively which can be parameterised by a piecewise continuously differentiable function and let be the region in enclosed by . Let be continuously differentiable. Then
Let . As we have seen before, we can write the above formula also as
Verify Green’s theorem in normal and tangent form for the vector field and the curve for
Let be a simple closed curve and let denote the region enclosed by . Using Green’s theorem, find a line integral along which yields the area of .
Find the area of the region bounded by the curve where
Let be a circle with radius centered at and oriented counterclockwise. Calculate
Note that where
Let and let be the boundary of which is oriented counterclockwise. Calculate
For an application of Green’s theorem see here.