Math Help

Green’s Theorem Click here for discussion of Topology of Curves and Domains

In a Nut Shell: Green's theorem gives the relationship between a line integral around a

simple closed curve, C, and a double integral over the plane region R bounded by C.

It is a special two-dimensional case of the more general Stokes' theorem.

Green’s theorem expressed in its standard form is

∫ P dx + Q dy = ∫ ∫ [ ∂Q/∂x ˗ ∂P/∂y ] dA

C R

where C is a curve enclosing the region, R, with element

of area dA. The curve, C, is said to be positively oriented

when traveling counterclockwise around C keeping the

region, R, enclosed to the left.

Note: The partial derivatives must be continuous throughout R else you will need

to modify the region to avoid discontinuities such as in a region R that is not

simply connected. i.e. A region that contains a hole.

Green’s theorem can also be expressed in its “curl form”.

F = P(x,y) i + Q(x,y) j and dr = dx i + dy j

So ∫ F . dr = ∫ F . T ds = ∫ Pdx + Q dy

C C C

where T is the unit tangential vector to the curve, C, n is the unit normal vector to the

curve and ds is the arc length along the curve. See the figure above.

i j k

Now curl F = ∂/∂x ∂/∂y ∂/∂z = [∂Q/∂x - ∂P/∂y] k

P(x,y) Q(x,y) 0

So ∫ F . dr = ∫ F . T ds = ∫ ∫ curl_z F dA (curl form of Green’s Theorem)

C C R

where curl_z F is the z-component of curl F = curl F . k

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