# What the hell is Green's third identity talking about?

• quasar987
In summary, Green's third identity is a fundamental theorem in vector calculus that relates a line integral around a closed curve to a double integral over the region enclosed by the curve. It is commonly used in physics and engineering to solve problems involving scalar and vector fields, such as in the study of fluid flow, electromagnetism, and heat transfer. This identity is also related to other mathematical theorems and has practical applications in various fields, including fluid dynamics, heat transfer, and electromagnetism. It is also useful in the study of differential equations and complex analysis.
quasar987
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Here it is on Mathworld:

http://mathworld.wolfram.com/GreensIdentities.html

On Wikipedia, it's a total carnage:

http://en.wikipedia.org/wiki/Green's_identities

In Mathworld, what are r, n and ds in cartesian coordinate?

r is the positive distance from the singularity, ds is the line element, and $$\frac{\partial{u}}{\partial{n}}\equiv\nabla{u}\cdot\vec{n}$$, i.e, the component of the gradient of u along the normal to the curve.

## 1. What is Green's third identity?

Green's third identity, also known as the third Green's theorem, is a mathematical concept that relates a line integral around a simple closed curve to a double integral over the region enclosed by the curve. It is a fundamental theorem in vector calculus.

## 2. How is Green's third identity used in science?

Green's third identity is commonly used in physics and engineering to solve problems involving scalar and vector fields, such as in the study of fluid flow, electromagnetism, and heat transfer. It provides a powerful tool for evaluating line and surface integrals.

## 3. Can you provide an example of Green's third identity in use?

One example of Green's third identity in use is in the calculation of the work done by a conservative force on a particle moving along a closed path. By applying the identity, the line integral of the force can be converted to a double integral of the force's potential over the enclosed region, which is often easier to evaluate.

## 4. Is Green's third identity related to other mathematical theorems?

Yes, Green's third identity is related to other theorems such as Green's first and second identities, which also involve the relationship between line and surface integrals. It is also closely linked to the fundamental theorem of calculus and the divergence theorem.

## 5. What are the practical applications of Green's third identity?

Aside from its use in solving problems in physics and engineering, Green's third identity has practical applications in fields such as fluid dynamics, heat transfer, and electromagnetism. It also has applications in other areas of mathematics, such as in the study of differential equations and complex analysis.

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