State and prove Green’s theorem.

In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It is the two-dimensional special case of Stokes' theorem

What is Green’s Theorem?

Green’s theorem is one of the four fundamental theorems of calculus, in which all of four are closely related to each other. Once you learn about the concept of the line integral and surface integral, you will come to know how Stokes theorem is based on the principle of linking the macroscopic and microscopic circulations. Similarly, Green’s theorem defines the relationship between the macroscopic circulation of curve C and the sum of the microscopic circulation that is inside the curve C.

Green’s Theorem Statement

Let C be the positively oriented, smooth, and simple closed curve in a plane, and D be the region bounded by the C. If L and M are the functions of x,y defined on the open region, containing D and have continuous partial derivatives, then the Green’s theorem is stated as

Where the path integral is traversed counterclockwise along with C.

Proof : -

The following is a proof of half of the theorem for the simplified area D, a type I region where C1 and C3 are curves connected by vertical lines (possibly of zero length). A similar proof exists for the other half of the theorem when D is a type II region where C2 and C4 are curves connected by horizontal lines (again, possibly of zero length). Putting these two parts together, the theorem is thus proven for regions of type III (defined as regions which are both type I and type II). The general case can then be deduced from this special case by decomposing D into a set of type III regions.

If it can be shown that if

...........(1)

and

...........(2)

are true, then Green's theorem follows immediately for the region D. We can prove (1) easily for regions of type I, and (2) for regions of type II. Green's theorem then follows for regions of type III.

Assume region D is a type I region and can thus be characterized, as pictured on the right, by

 D = { (x,y) | a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x) }

where g1 and g2 are continuous functions on [a, b]. Compute the double integral in (1):

.................(3)

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 (anticlockwise). On C₂ and C₄, x remains constant, meaning

Therefore,

..........(4)

Combining (3) with (4), we get (1) for regions of type I. A similar treatment yields (2) for regions of type II. Putting the two together, we get the result for regions of type III.