# Help in proof (Green's Theorem)

In summary, Green's Theorem is a tool used in vector calculus to calculate line integrals over closed curves. It is often used in proofs involving line integrals and can also be applied to prove properties of vector fields. The conditions for using Green's Theorem include a closed curve and a continuously differentiable vector field. It can only be applied to two-dimensional vector fields, but there are similar theorems for three-dimensional fields. Some real-world applications of Green's Theorem include calculating work and fluid flow, as well as studying electromagnetism.

## Homework Statement

I have a doubt in proving the attached theorem. I have found that divergence theorem can be applied. However I am not able to arrive at the exact equation.

Attached

## The Attempt at a Solution

Derived the LHS. Couldn't proceed from there.

Last edited:
I consider u to be u1 i + u2 j and found the divergence. How to proceed from there?

## 1. What is Green's Theorem?

Green's Theorem is a mathematical tool used in vector calculus to calculate the line integral of a two-dimensional vector field over a closed curve. It establishes a relationship between line integrals and double integrals, allowing for easier and more efficient calculations.

## 2. How is Green's Theorem used in proofs?

Green's Theorem is often used in proofs involving the calculation of line integrals over closed curves. It can also be used to prove various properties of vector fields, such as the divergence theorem and the curl theorem.

## 3. What are the conditions for using Green's Theorem?

There are two main conditions for using Green's Theorem: the curve over which the line integral is being calculated must be a closed curve, and the vector field being integrated must be continuously differentiable within the region enclosed by the curve.

## 4. Can Green's Theorem be applied to three-dimensional vector fields?

No, Green's Theorem can only be applied to two-dimensional vector fields. However, there are similar theorems for three-dimensional vector fields, such as Stokes' Theorem and the Divergence Theorem.

## 5. What are some real-world applications of Green's Theorem?

Green's Theorem has various applications in physics, engineering, and other fields. For example, it can be used to calculate the work done by a force on a moving object, or to determine the flow of fluid through a closed curve. It is also used in the study of electromagnetism to calculate the circulation of a magnetic field around a closed path.

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