Green’s Theorem – Statement, Proof, Uses & Solved Examples

Green’s Theorem is a helpful rule in mathematics that connects a line integral around a closed curve with a double integral over the region enclosed by that curve. It applies to two-dimensional vector fields.

In simple terms, Green’s Theorem says that instead of calculating a line integral around the boundary (like walking around the edge of a field), you can get the same result by doing a double integral over the entire area inside the boundary (like measuring everything within the field). This makes calculations much easier in many cases.

This theorem works in the plane (2D) and is especially useful in physics and engineering problems, like calculating work done by a force field or the circulation and flow of fluids.

Green’s Theorem is also a special case of a more general rule called Stokes' Theorem, which works in three or more dimensions.

Think of it like this: if a field has some rotation or circulation happening across a surface, Green’s Theorem lets you measure all of that rotation by just going around the edge of the surface. It turns the complicated idea of measuring everything inside into a simpler path around the outside.

• It means that at a point (x,y) there are tiny closed curves around the center.
• A microscopic circulation is a bunch of tiny curves where each of the tiny curves is the tendency of the vector field circulating around that location.
• Green’s theorem establishes a macroscopic relationship around the curve “C” that encloses the boundary of area “D”

• In simpler words, Green’s theorem states that if we add up all the microscopic curves in an area “D” then that total would be equal to the macroscopic circulation over the same area within the curve “C”.

What is Green’s Theorem?

Green’s Theorem says that instead of solving a line integral around the edge of a flat shape, you can solve a double integral over the area inside it. This makes certain problems easier to solve. It connects the boundary of a shape with the region inside it. Green’s Theorem is used in vector fields and helps in calculating area, flow, or circulation in 2D space.

where

• C is a smooth curve along a closed path, D is the region bounded by curve “C”
• F1 and F2 are the functions of (x,y) for region “D” which have continuous partial derivatives.

Consider that “C” is a simple curve that is positively oriented along region “D”.

The functions M and N are defined by (x,y) within the enclosed region “D”, which has continuous partial derivatives.

Green’s theorem states that,

We will prove Green’s theorem in 3 phases:

1. It is applicable to the curves for the limits between x = a to x = b.
2. For curves that are bounded by y = c and y = d.
3. For the curves that are similar to the above conditions.

Step 1 :

Step 2 : =

The double integrals for the same region,

Step 3 :

Step 4 : – equation (1)

From this, we have confirmed that Green’s theorem is applicable to the curves for limits between x = a to x = b.

Similarly to prove Green’s theorem for the curves bounded by y = c and y = d,

Step 1 :

Step 2 :

Step 3 :

Step 4 : – equation (2)

Combining Equations (1) and (2),

Step 5 :

Step 6 :

Even when we are working with several curves in an enclosed region this theorem is still valid as we will simplify using Green’s theorem for each type of curve.

Learn about Rolle’s Theorem

Area of Curve using Green’s Theorem

If we are in a two-dimensional simple closed curve and F(x,y) is defined everywhere inside Curve “C”, we will use Green’s theorem to convert the line integral into double form.

The area of region “D” is equal to the double integral of f(x,y) = 1 dA

Area of D =

If f(x,y) = 1,

Then

In general,

If “C” is in a counter-clockwise direction which is a simple closed curve that bounds a region,

Area of region D is bounded by curve “C” =

Area of D =

Where (x,y) = .

Difference Between Green’s Theorem and Stokes Theorem

Green’s Theorem	Stokes Theorem
Green’s theorem relates a double integral over a plane region “D” to a line integral around its curve.	It relates the surface integral over surface “S” to a line integral around the boundary of the curve of “S” (which is the space boundary).
Green’s theorem talks about only positive orientation of the curve.	Stokes theorem talks about positive and negative surface orientation.
Green’s theorem is a special case of stoke’s theorem in two-dimensional space.	Stokes theorem is generally used for higher-order functions in a three-dimensional space.
Is the function of Green’s theorem, over area dA, enclosed by a boundary curve “C”.	is the statement of stokes theorem, over any surface “S”.

Uses of Green’s Theorem

The following are the uses of Green’s theorem

1. Green’s theorem converts a line integral to a double integral over microscopic circulation in a region.
2. It is applicable only over closed paths.
3. It is used to calculate the vector fields in a two-dimensional space.
4. It is also used to calculate the area and tangent vector of a boundary oriented in an anticlockwise direction.

Applications of Green’s Theorem 

Green’s Theorem is useful in many areas like physics, engineering, and maths, especially when dealing with shapes and flows in 2D. Here are some simple examples of how it's used:

1. Work Done by a Force: When an object moves along a closed path in a force field, Green’s Theorem helps to calculate the total work done. Instead of checking each point along the path, you can use the area inside the path to find the result more easily.
2. Flow of Fluids: If a liquid is moving in a flat region, Green’s Theorem can be used to find out how much of it is spinning (circulating) or flowing out through the boundary. This is helpful in studying how fluids move in pipes, rivers, or air systems.
3. Finding Area Inside a Curve: Green’s Theorem can also be used as a shortcut to find the area inside a loop or boundary. By using the right setup in the formula, it changes a complicated path problem into a simple area calculation.

Solved Examples of Green’s Theorem

Example 1. Calculate the line integral where “c” is a rectangle and its vertices are (1,1) , (4,1) , (4,5) , (1,5).

Solution: Let F(x,y) =[ P(x,y), Q(x,y)], where P and Q are the two functions.

=

Then,

Hence,

“D” is the rectangular region enclosed by the curve “C”.

By Green’s theorem,

.

= -84

The line integral of the given function is -84

Example 2. Calculate the work done on a particle by a force field

The particle traverses the circle in an anti-clockwise direction the start and the endpoints are (2,0).

Solution: “C” is the circle and “D” is the region enclosed by the circle “C”.

The work done on the particle is,

Using Green’s theorem,

Let F(x,y) =[ P(x,y), Q(x,y)], where P and Q are the two functions.

=

By Green’s theorem,

= -2 (area of “D”)

=

Hope this article was informative and helpful for your studies and exam preparations. Stay tuned to the Testbook app for more updates and topics related to Mathematics and various such subjects. Also, reach out to the test series available to examine your knowledge regarding related exams.

If you are checking Green’s Theorem article, also check related maths articles:
Alternate interior angles	Cumulative frequency distribution
Exterior Angle Theorem	Semi circle
Derivative of sin 3x	Apollonius Theorem

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