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).

Read more about this topic:  Green's Theorem

Famous quotes containing the words proof, simple and/or region:

    “There is no better proof of a man’s being truly good than his desiring to be constantly under the observation of good men.”
    —François, Duc De La Rochefoucauld (1613–1680)

    “Your apple face, the simple crèche
    Of your arms, the August smells
    Of your skin. Then I sorted your clothes
    And the loves you had left, Elizabeth,
    Elizabeth, until you were gone.”
    —Anne Sexton (1928–1974)

    “I was with Hercules and Cadmus once,
    When in a wood of Crete they bayed the bear
    With hounds of Sparta: never did I hear
    Such gallant chiding; for besides the groves,
    The skies, the fountains, every region near
    Seemed all one mutual cry. I never heard
    So musical a discord, such sweet thunder.”
    —William Shakespeare (1564–1616)