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.