Table of Content

• What is Green’s Theorem?
• Green’s Theorem Statement
• Green’s Theorem Proof
• Green’s Theorem: Closed Curved Area
• Difference Between Green’s Theorem and Strokes Theorem
• Green’s Theorem Example
• Green’s Theorem Applications

Feature	Green’s Theorem	Stokes’ Theorem
Context	Applies in the plane (2 dimensions)	Applies in three-dimensional space (3 dimensions).
Type of Integral	Relates a line integral around a closed curve to a double integral over the region it encloses.	Relates a line integral of a vector field around a closed curve to a surface integral of the curl of the vector field over the surface of its bounds.
Dimension	2-dimensional theorem.	3-dimensional theorem
Field Type	Typically applied to conservative vector fields (where curl is zero).	Generally applied to any sufficiently smooth vector field.
Formal Statement	∮C​(Pdx+Qdy) = ∬R​(∂x/∂Q​−∂y/∂P​)dA	∮C​F⋅dr=∬S​(∇×F)⋅dS
Fields Involved	Involves a vector field F = (P, Q)	Involves a vector field F = (F1, F2, F3)
Curl Condition	Does not require the vector field to have a non-zero curl.	Requires the vector field to have a well-defined curl (non-zero in some regions).
Application Example	Calculation of area enclosed by a curve in the plane.	Calculation of flux through a surface in three-dimensional space.
Physical Interpretation	Relates circulation (or line integral) to area (or double integral).	Relates circulation around a curve to flux through a surface.

Green’s Theorem Example

Example: Use Green’s Theorem to calculate the work done by a force field F(x, y) = (y², 2xy) around a closed rectangular path in the xy-plane.

Solution:

Step 1: Define Path and Force Field

We’re given a force field F(x, y) = (y², 2xy) and a rectangular path. Let’s define the rectangle with:

• Starting point: (x₁, y₁) = (1, 1)
• Ending point: (x₂, y₂) = (2, 3)

This creates a rectangle with side lengths of 1 unit (x₂ – x₁) and 2 units (y₂ – y₁). We’ll assume a counter-clockwise direction around the rectangle.

Step 2: Verify Green’s Theorem Requirements

• Path is a simple, closed curve (rectangle).
• It encloses a simply connected region (no “holes” inside the rectangle).
• Force field F(x, y) has continuous partial derivatives throughout the rectangle (y² and 2xy are both continuous everywhere).

Therefore, Green’s theorem can be applied to this scenario.

Step 3: Apply Green’s Theorem

We can either evaluate the line integral directly or use Green’s theorem to convert it into a double integral. Here, we’ll use Green’s theorem:

∮_C F(x, y) • dr = ∫∫_D (∂Q/∂x – ∂P/∂y) dA

Step 4: Find Partial Derivatives

We need ∂Q/∂x and ∂P/∂y:

• ∂Q/∂x = 2y (partial derivative of 2xy w.r.t. x)

• ∂P/∂y = 2y (partial derivative of y^2 w.r.t. y)

Step 5: Convert Line Integral to Double Integral

Plugging the partial derivatives into Green’s theorem:

∫∫_D (2y – 2y).dA = 0.dA

Since ∂Q/∂x – ∂P/∂y cancels out, the double integral becomes zero across the entire area.

Step 6: Interpretation

Green’s theorem tells us that in this case, the work done by the force field F around the closed rectangular path is zero.

Green’s Theorem Applications

Green’s theorem has various applications in physics, engineering, and mathematics. Here are some key examples:

• Calculating Work Done by Forces: Imagine a force field acting on an object along a closed path. Green’s theorem can be used to relate the line integral of the force (work done) around the path to the double integral of a quantity related to the circulation of the force field within the enclosed area.

• Fluid Flow: Green’s theorem can be applied to analyze two-dimensional fluid flow. By considering a velocity field representing the fluid’s motion, the theorem can help calculate the net circulation of the fluid around a closed curve or the total outward flux across the curve.

• Area Enclosed by a Curve: In a simpler form, Green’s theorem can be used to calculate the area enclosed by a simple closed curve. By setting one of the component functions in the line integral to zero and using a specific circulation property, the double integral reduces to calculating the area.

Green’s Theorem Problems and Solutions

Problem 1: Calculating Work Done: A force field F(x, y) = (2xy, y²) acts in the xy-plane. Calculate the work done by this force field along a positively oriented square path with a of side length 1 unit, starting at the point (0, 0).

Solution:

∫_C F • dr

where C is the square path. Green’s theorem allows us to convert this into a double integral:

∫∫_D (∂Q/∂x – ∂P/∂y) dA

where D is the area enclosed by the square path (unit square), P(x, y) = 2xy (from F’s first component), Q(x, y) = y² (from F’s second component), and dA is the area element.

Find Partial Derivatives:

∂Q/∂x = 2y (partial derivative of y² w.r.t. x) ∂P/∂y = 2x (partial derivative of 2xy w.r.t. y)

Evaluate Double Integral:

∫∫_D (2y – 2x) dA

Since the square path is from (0, 0) to (1, 1) with constant side length, we can directly evaluate the double integral:

[Tex]∫_0^1 [/Tex][Tex]∫_0^1[/Tex] (2y – 2x) dy dx = 0

Interpretation:

Work done by the force field around the closed square path is zero. This could be because the force field is conservative in this scenario, or because the square path itself creates cancelling effects along opposite sides.

Problem 2: Findin Area Enclosed by a Curve: Find the area enclosed by the curve C defined by x^2 + y^2 = 4 (circle centred at the origin with radius 2) using Green’s theorem.

Solution:

Green’s Theorem for Area: We can use a special case of Green’s theorem where one function in the line integral is zero. Here, we set P(x, y) = 0 and Q(x, y) = x. The double integral then becomes:

∫∫_D ∂Q/∂x dA = ∫∫_D dA

which essentially calculates the area enclosed by the curve C.

Line Integral and Double Integral: The line integral becomes:

∫_C 0 dx + x dy

Partial Derivative:

∂Q/∂x = 1 (partial derivative of x w.r.t. x)

Evaluate Double Integral:

We can use the standard formula for the area of a circle (πr^2) since Green’s theorem essentially converts the line integral into calculating the area:

∫∫_D dA = π × 2² = 4π

Interpretation:

Area enclosed by the circle x² + y² = 4 is 4π square units.

Problem 3: Fluid Flow: A two-dimensional fluid flow has a velocity field F(x, y) = (y, -2x). Consider a rectangular path in the xy-plane with vertices at (0, 0), (1, 0), (1, 2), and (0, 2).

• (a) Calculate the circulation of the fluid around the rectangular path using Green’s theorem.
• (b) Interpret the result in terms of the fluid flow.

Solution:

(a) We can directly apply Green’s theorem. Here’s the breakdown:

• P(x, y) = y (from F’s first component)
• Q(x, y) = -2x (from F’s second component)

We need to find ∂Q/∂x – ∂P/∂y = -2 – 1 = -3

Green’s theorem states:

∫_C F • dr = ∫∫_D (-3) dA

where C is the rectangular path and D is the area it encloses.

Since the path is a rectangle with constant side lengths, we can directly evaluate the double integral:

[Tex]∫_0^1 [/Tex][Tex]∫_0^2[/Tex] (-3) dy.dx = -6

Interpretation:

(b) Negative value (-6) for the circulation indicates that the fluid has a net counter-clockwise flow around the rectangle.

The circulation represents the total “turning tendency” of the fluid along the closed path. A negative value signifies counter-clockwise circulation in this context.

Also Read:

• Vector Calculus
• Divergence theorem
• Application of Gauss Theorem

FAQs on Green’s Theorem

What is the Use of Green’s Theorem?

Green’s Theorem is commonly used for the integration of lines when combined with a curved plane. It is used to integrate the derivatives in a plane. If the line integral is given, it is converted into the surface integral or the double integral or vice versa with the help of this theorem.

Who Invented the Green Theorem?

It is named after George Green, who stated a similar result in an 1828 paper titled An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. In 1846, Augustin-Louis Cauchy published a paper stating Green’s theorem as the penultimate sentence.

Where is Green’s theorem used in real life?

Some of applications of Greens’s Theorem include:

• Calculating the area of irregular shapes.
• Determining the circulation and flux of fluid across a surface in engineering tasks.
• Facilitating the analysis of electromagnetism field flux in physics.

What are Both Forms of Green’s Theorem?

There are two primary forms of Green’s Theorem:

• Circulation Form
• Flux Form

What is the Unit of Green’s Function?

The unit of a Green’s function depends on the context in which it is used and the specific type of differential equation it is associated with. Generally, Green’s functions are used to solve linear differential equations, and their units are determined by the differential operator they are associated with and the nature of the source term.