Statement of Green’s theorem
Green’s theorem is the extension of Stoke’s theorem and the divergence theorem. According to this theorem, if ϕ and ψ be the scalar functions, then
![State and prove green's theorem](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/08/1.jpg?resize=567%2C107&ssl=1)
Proof of Green’s theorem
Let a vector
![](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/08/2.jpg?resize=286%2C61&ssl=1)
According to the divergence theorem.
![](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/08/3.jpg?resize=299%2C102&ssl=1)
![](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/08/4.jpg?resize=718%2C104&ssl=1)
![](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/08/5.jpg?resize=528%2C60&ssl=1)
![](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/08/6-1.jpg?resize=597%2C43&ssl=1)
![](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/08/7-1.jpg?resize=391%2C70&ssl=1)
Substituting the value of eqn. (2) in eqn. (1) we get
![](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/08/8-1.jpg?resize=564%2C104&ssl=1)
![State and prove green's theorem](https://i0.wp.com/physicswave.com/wp-content/uploads/2021/08/9.jpg?resize=586%2C119&ssl=1)
Which is the Green’s Theorem.