Green's Theorem

1. Setting

[pic 1]

Let a closed curve C be set on x,y plane.

Domain D is covered by closed curve C.

[pic 2]

Let's functions, M(x,y) and N(x,y) exist and they have partial derivatives on D.

The formula can be notated like it in [pic 2] on C.

[pic 3]

This formula can be divided like [pic 3] 

because M(x,y)dx and N(x,y)dy are on same closed curve C. 

2. Induction

2-1. Considering The Formula at X Coordinate

First, we consider the formula in [pic 2] at x coordinate.

[pic 4]

[pic 5]

We set a domain like a formula in [pic 5].

[pic 6]

So, result is like [pic 6] on x coordinate.

2-2. Considering The Formula at Y Coordinate

Next, we also consider the formula in [pic 2] at y coordinate.

[pic 7]

[pic 8]

We set a domain like a formula in [pic 8].

[pic 9]

So, result is like [pic 6] on y coordinate.

3. Conclusion

[pic 10]

We can combine same domain.

Therefore, we can prove Green's Theorem.