GREEN’S THEOREMNS ZAIN JAVEDNS IQRA NAWAZISH

GREEN’S THEOREM

•DEFINITION & PROOF•RELATION TO OTHER THEOREMS

•APPLICATIONS

THE IDEA OF GREEN’S THEROEM• When C is an oriented closed path (i.e., a path where the

endpoint is the same as the beginning point), the integral • ∫ CF ⋅ ds represents the circulation of F around C. If F were

the velocity field of water flow, for example, this integral would indicate how much the water tends to circulate around the path in the direction of its orientation.

• One way to compute this circulation is, of course, to compute the line integral directly. But, if our line integral happens to be in two dimensions (i.e., F is a two-dimensional vector field and C is a closed path that lies in the plane), then Green’s theorem applies and we can use Green’s theorem as an alternative way to calculate the line integral.

Green’s theorem is simply a relationship between the macroscopic circulation around the curve C and the sum of all the microscopic circulation that is inside C. If C is a closed curve in the plane, then it surrounds some region D in the plane. D is the “interior” of the curve C. Green’s theorem says that if you add up all the microscopic circulation inside C, then that total is exactly the same as the macroscopic circulation around C

• “Adding up” the microscopic circulation in D means taking the double integral of the microscopic circulation over D. Therefore, we can write Green’s theorem as

PROOF OF GREEN’S THEOREM

AREA OF A REGION• Green's theorem can be used to compute area by line

integral. The area of D is given by

Provided we choose L and M such that:

Then the area is given by:

Possible formulas for the area of D include

RELATIONSHIP WITH OTHER THEROEMS

Green's theorem is a special case of Stoke’s Theorem, when applied to a region on the xy-plane:

Considering only two-dimensional vector fields, Green's theorem is equivalent to the following two-dimensional version of the divergence theorem:

DISCRETE GREEN’S THEOREM• In discrete geometry, discrete versions of Green’s

Theorem describe the relationship between a double integral of a function over a generalized rectangular domain D (a domain which is formed by a finite unification of rectangles), and a linear combination of the given function's cumulative distribution function at the corners of the domain.

• The discrete Green's theorem is being used for fast calculations in computer applications such as objects detection in images, and in efficient calculations of probabilities.

METHOD OF IMAGE CHARGES

• A method used in electrostatics takes advantage of the uniqueness theorem (derived from Green's theorem).

• The method of image charges (also known as the method of images and method of mirror charges) is a basic problem-solving tool in electrostatics. The name originates from the replacement of certain elements in the original layout with imaginary charges, which replicates the boundary conditions of the problem

TWO-DIMENSIONAL IMAGE PROCESSING USING GREEN’S THEOREM

• Its basically done in series of complicated step. One of which is the estimation of area of the image using green’s theorem.

• Green’s theorem is a vector identity that relates the surface integral of a boundary to its line integral.  It says that the area of a surface can be calculated by the equation:

which can be expressed in discrete form as:

• When dealing with images, we can use the coordinates (x and y) or the pixel locations of the image in order to compute for the area of interest.  We begin by applying these basic methods on the simple figures below:

We first make sure the images are in their binary form in order for the machine to easily interpret the values. The area to calculate should be set to white, and the background black. Then we count the number of these figures which precisely calculates the area of the figure.

This is followed by some godlike programming :P which ultimately in the end gives us sharp images that are less blur.

• It’s used in finger prints enhancement and analysis.

Fingerprint ridge enhancement. A. a grayscale of the original fingerprint; B. an enhanced image of the fingerprint after filtering.

Binary representation of the land area.

Helicopter view of the National Science Complex of the University of the Philippines Diliman.

Applications of Green's Theorem to Overmoded Coaxial Waveguide

• A waveguide is a structure which guides waves, such as electromagnetic waves or sound waves.

• Coaxial cable, or coax, is an electrical cable with an inner conductor surrounded by a flexible, tubular insulating layer, surrounded by a tubular conducting shield. The term coaxial comes from the inner conductor and the outer shield sharing the same geometric axis.

• Electromagnetic boundary-value problems supported by closed-form solutions provide a framework for efficient and optimized design methods. Closed-form Green's theorem models are used when a well-defined source is coupled to a well-defined space or structure. Here the Green's theorem is applied to describe the relationship between the current in a coaxial probe and the electromagnetic fields inside an overmoded coaxial airspace .

Geometry and field plot for coaxial wrap-around TE01 mode converter

Taper Converters

The Application of Green's Theorem to the Solution of Boundary-Value Problems in Linearized Supersonic Wing Theory

• With a recent trend of the world wide growth of air transportation, development of a next generation supersonic transport (SST) is under consideration in the United States, Europe, and Japan. There have been a few supersonic transport so far, such as, the TU-144 developed by the Soviet Union and the Concorde by the joint of UK and France. But the TU-144 ceased its regular operation some time before 1985 because of problems with the engines and wing design.

• Now these problems were raised as general methods of solution are given for the two and three dimensional steady-state and two-dimensional unsteady-state equations. In the absence of thickness effects, linear theory yields solutions consistent with the assumptions made when applied to lifting-surface problems for swept-back plan forms at sonic speeds. The solutions of the particular equations are determined in all cases by means of Green's theorem.

• Anyhow these designs are still on paper and only in the form theories and abstract published by the Tohoku University, Sendai, Japan and National Aerospace Laboratory, Chofu, Tokyo 182, Japan

AREA OF REGION WITH HOLES

• Unlike other line integral formulas Green’s Theorem can be used to find the area of regions with holes in them. Consider the following example. The region is not simply connected.

MOMENTS & CENTROIDS• Green’s theorem allows to express the

coordinates of the centroid as line integrals. One just has to find the right vector fields for each coordinate. For example,

PLANIMETER• A planimeter is a measuring instrument used to

determine the area of an arbitrary two-dimensional shape.• The main types of mechanical planimeter are polar, linear

and Prytz or "hatchet" planimeters.• They consist of a linkage with a pointer on one end, used

to trace around the boundary of the shape. The other end of the linkage is fixed for a polar planimeter and restricted to a line for a linear planimeter.

• Tracing around the perimeter of a surface induces a movement in another part of the instrument and a reading of this is used to establish the area of the shape.

• The planimeter contains a measuring wheel that rolls along the drawing as the operator traces the contour.

Polar planimeter

A linear planimeter on scrolls for the determination of stretched shapes

• The area of the shape is proportional to the number of turns through which the measuring wheel rotates when the planimeter is traced along the complete perimeter of the shape.

Linear planimeter

Polar planimeter

For the linear planimeter the movement of the "elbow" E is restricted to the y-axis. For the polar planimeter the "elbow" is connected to an arm with fixed other endpoint O. Connected to the arm ME is the measuring wheel with its axis of rotation parallel to ME. A movement of the arm ME can be decomposed into a movement perpendicular to ME, causing the wheel to rotate, and a movement parallel to ME, causing the wheel to skid, with no contribution to its reading.

• The operation of a linear planimeter can be justified by applying Green’s Theorem onto the components of the vector field N, given by:

where b is the y-coordinate of the elbow E.This vector field is perpendicular to the measuring arm EM:

and has a constant size, equal to the length m of the measuring arm:

because:

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