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1. Introduction

In physics and mathematics, Green'Green’s theorem gives indicates the relationship between a line integral around a simple closed curve C and a double integral over the planeplane region D bounded by C. This theorem is an application of the fundamental Theoremtheorem of calculus for to integrating a certain combinations of derivatives over a plane. It This theorem can be proven easily proven for rectangular and triangular regions. As Bothboth sides of it’sits equality are finitely additive and almost all planar regions can be divided into triangles and rectangles, so that the result holds for any planar- region practically all of, which can be divided in to triangles and rectangles.  1
This proves the theorem for reasonably shaped regions.2 It’s
Its generalization to the non planar surfaces ( proved directly proved from it by using the finite additivity of both sides ) is the Stokes Theorem theorem described below.

1.1 GreensGreen’s Theorem

Its The formal statement of Green’s theorem3 is as follows : : Let S be a sufficiently nice region in the plane, and let S be its boundary;then. Then, we have;

,

where the boundary, S is traversed counterclock wise on it’sits outside cycle, (and clockwise on any internal cycles as you can be verify verified using zippers.)).

Meaning of this tTheorem interpretation: 4 Green'sGreen’s theorem is a form that the fundamental theorem of calculus taketakes in the context of integrals over planar regions.

For a rectangle: By Using the ordinary fundamental theorem of calculus, we have;

.

For a right triangle: fFor convenience, we choose a triangle bounded by line x = 0, y = 0, and .

We similarly getobtain:

.

Rearrangement of right hand side gives the Theorem for rectangles and right triangles is obtained by rearranging the right hand side of the equation.
It means thatThus,, for R, a rectangle or right triangle in the x-y plane, (for which dS = dSk), we have

.

Both sides of this equation is finiteare finitely additive:, that isi.e., if we evaluate either side overwe take two disjoint regions, and evaluate either one over both, you get the result will be equal to the sum of their valuesthe result of separate evaluations on the two regions separate .. This is true even if the regions share a common boundary , 5 because the line integrals will cancel out over the common boundary which that ceases to be a  6 boundary.
The result follows from additivity for any region that can be broken updivided into rectangles and triangles, which accounts for most regions we will encounter.

Explanations

1. Introduction

In physics and mathematics, Green'Green’s theorem gives indicates the relationship between a line integral around a simple closed curve C and a double integral over the planeplane region D bounded by C. This theorem is an  1 application of the fundamental Theoremtheorem of calculus for to integrating a certain combinations of derivatives over a plane. It This theorem can be proven easily proven for rectangular and triangular regions. As Bothboth sides of it’sits equality are finitely additive and almost all planar regions can be divided into triangles and rectangles,  2 so that the result holds for any planar- region practically all of, which can be divided in to triangles and rectangles.
This proves the theorem for reasonably shaped regions. It’s
Its generalization to the non planar surfaces ( proved directly proved from it by using the finite additivity of both sides ) is the Stokes Theorem theorem described below.

1.1 GreensGreen’s Theorem

Its The formal statement of Green’s theorem3 is as follows : : Let S be a sufficiently nice region in the plane, and let S be its boundary;then. Then, we have;

,

where the boundary, S is traversed counterclock wise on it’sits outside cycle, (and clockwise on any internal cycles as you can be verify verified using zippers.)).

Meaning of this tTheorem interpretation: 4 Green'sGreen’s theorem is a form that the fundamental theorem of calculus taketakes in the context of integrals over planar regions.

For a rectangle: By Using the ordinary fundamental theorem of calculus, we have;

.

For a right triangle: fFor convenience, we choose a triangle bounded by line x = 0, y = 0, and .

We similarly get : >5

.

Rearrangement of right hand side gives the Theorem for rectangles and right triangles is obtained by rearranging the right hand side of the equation.
It means thatThus,, for R, a rectangle or right triangle in the x-y plane, (for which dS = dSk), we have

.

Both sides of this equation is finiteare finitely additive:, that isi.e., if we evaluate either side overwe take two disjoint regions, and evaluate either one over both, you get the result will be equal to the sum of their valuesthe result of separate evaluations on the two regions separate .. This is true even if the regions share a common boundary , because the line integrals will cancel out over the common boundary which that ceases to be a   boundary.
The result follows from additivity for any region that can be broken updivided into rectangles and triangles, which accounts for most regions we will encounter.

Explanations

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