Green's Theorem - Proof When D Is A Simple Region

Proof When D Is A Simple Region

The following is a proof of the theorem for the simplified area D, a type I region where C2 and C4 are vertical lines. A similar proof exists for when D is a type II region where C1 and C3 are straight lines. The general case can be deduced from this special case by approximating the domain D by a union of simple domains.

If it can be shown that

and

are true, then Green's theorem is proven in the first case.

Define the type I region D as pictured on the right by

where g1 and g2 are continuous functions on . Compute the double integral in (1):

\begin{align} \iint_D \frac{\partial L}{\partial y}\, dA & =\int_a^b\,\int_{g_1(x)}^{g_2(x)} \frac{\partial L}{\partial y} (x,y)\,dy\,dx \\ & = \int_a^b \Big\{L(x,g_2(x)) - L(x,g_1(x)) \Big\} \, dx.\qquad\mathrm{(3)} \end{align}

Now compute the line integral in (1). C can be rewritten as the union of four curves: C1, C2, C3, C4.

With C1, use the parametric equations: x = x, y = g1(x), a ≤ x ≤ b. Then

With C3, use the parametric equations: x = x, y = g2(x), a ≤ x ≤ b. Then

The integral over C3 is negated because it goes in the negative direction from b to a, as C is oriented positively (counterclockwise). On C2 and C4, x remains constant, meaning

Therefore,

\begin{align} \int_{C} L\, dx & = \int_{C_1} L(x,y)\, dx + \int_{C_2} L(x,y)\, dx + \int_{C_3} L(x,y)\, dx + \int_{C_4} L(x,y)\, dx \\ & = -\int_a^b L(x,g_2(x))\, dx + \int_a^b L(x,g_1(x))\, dx.\qquad\mathrm{(4)} \end{align}

Combining (3) with (4), we get (1). Similar computations give (2).

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