# Green's theorem and area calculation

• Jhenrique
In summary, Green's theorem states that the area of a region enclosed by a curve C can be calculated by taking the line integral of a vector field around C. The choices of M and N in the formula are arbitrary, as long as they result in an equivalent double integral for the area. The example given in the conversation is one possible choice, but there are likely others that would also work.
Jhenrique
In wiki there is the follows formula:

https://en.wikipedia.org/wiki/Green's_theorem#Area_Calculation

But, I don't understand why M = x and L = -y. I don't found this step in anywhere.

Jhenrique said:
In wiki there is the follows formula:

https://en.wikipedia.org/wiki/Green's_theorem#Area_Calculation

But, I don't understand why M = x and L = -y. I don't found this step in anywhere.

Go back to the very definition of the Green's Theorem in the 'Theorem' section of the article. You pick M and N arbitrarily so that the equivalent double integral for the area of the region enclosed by the curve C can be expressed as a line integral around C. These choices of M and N are not necessarily unique.

1 person
Jhenrique said:
In wiki there is the follows formula:

https://en.wikipedia.org/wiki/Green's_theorem#Area_Calculation

But, I don't understand why M = x and L = -y. I don't found this step in anywhere.

If ##\vec F = \langle P(x,y),Q(x,y)\rangle## then Green's theorem is$$\iint_R Q_x - P_y~dA =\oint_C \vec F\cdot d\vec r$$If ##Q_x - P_y = 1## the double integral on the left gives the area of the enclosed region ##R##. Any choices of ##P## and ##Q## that give ##1## will work. The choices in your example are simple examples that work but you could likely find others.

## 1. What is Green's theorem and how is it used?

Green's theorem is a mathematical tool used to calculate the area of a two-dimensional region bounded by a simple closed curve. It relates the line integral over the curve to a double integral over the region, making it easier to calculate the area of irregular shapes.

## 2. How is Green's theorem derived?

Green's theorem is derived from the fundamental theorem of calculus. It states that the line integral of a vector field over a curve is equal to the double integral of the curl of the vector field over the region bounded by the curve.

## 3. What is the formula for calculating the area using Green's theorem?

The formula for calculating the area using Green's theorem is A = ∫∫(∂Q/∂x - ∂P/∂y) dA, where P and Q are the components of the vector field and dA is the infinitesimal area element.

## 4. Can Green's theorem be used for any shape?

Green's theorem can be used for any shape as long as it is bounded by a simple closed curve. This means that the curve does not intersect itself and has a well-defined inside and outside.

## 5. How is Green's theorem related to other theorems in mathematics?

Green's theorem is related to other theorems in mathematics such as Stokes' theorem and the Divergence theorem. These theorems are all based on the fundamental theorem of calculus and are used to relate different types of integrals in multivariable calculus.

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