The Green 's theorem is a calculation method used to connect line integrals double integrals or surface area. The functions involved must be denoted as vector fields and defined within the path C.
For example, a line integral expression can be very difficult to solve; however by implementing Green's theorem, double integrals become quite basic. It is always important to respect the positive direction of the trajectory, this refers to the anti-clockwise direction.
Green's theorem is a particular case of Stokes's theorem, where the projection of the vector function is carried out in the xy plane.
Definition
The expression of Green's Theorem is as follows:
The first term shows the line integral defined by the path “C”, of the scalar product between the vector function “F” and that of the vector “r”.
C: It is the defined path on which the vector function will be projected as long as it is defined for that plane.
F: Vector function, where each of its components is defined by a function as such (f, g).
r: It is a vector tangent to the region R over which the integral is defined. In this case we operate with a differential of this vector.
In the second term we see Green's theorem developed, where the double integral defined in the region R of the difference of the partial derivatives of g and f is observed, with respect to x and y respectively. By an area differential that is nothing more than the product of both two-dimensional differentials (dx.dy).
This theorem is perfectly applicable for space and surface integrals.
Demonstration
To prove Green's theorem in a simple way, this task will be broken down into 2 parts. First of all, we will assume that the vector function F only has a definition in the versor i. While the function "g" corresponding to the versor j will be equal to zero.
Author
F = f (x, y) i + g (x, y) j = f (x, y) i + 0
r = x i + y j
dr = dx i + dy j
First we develop the line integral over path C, for which the path has been sectorized into 2 sections that go first from a to b and then from b to a.
The definition of the fundamental theorem of calculus is applied for a definite integral.
The expression is rearranged into a single integral, the negative is made a common factor, and the order of the factors is reversed.
When observing this expression in detail, it becomes evident that when applying the primitive function criteria, we are in the presence of the integral of the expression derived from f with respect to y. Evaluated in parameters
Now it is enough to suppose that the vector function F is defined only for g (x, y) j. Where when operating in a manner similar to the previous case, the following is obtained:
To finish, the 2 proofs are taken and joined in the case where the vector function takes values for both versores. In this way, it is shown how the line integral after being defined and considered as a one-dimensional trajectory, can be fully developed for the plane and space.
F = f (x, y) i + g (x, y) j
In this way, Green's theorem is proved.
Applications
The applications of Green's theorem are wide in the branches of physics and mathematics. These extend to any application or use that can be given to line integration.
The mechanical work done by a force F through a path C, can be developed by a line integral that is expressed as a double integral of an area by Green's theorem.
The moments of inertia of many bodies subjected to external forces at different points of application also respond to line integrals that can be developed with Green's theorem.
This has multiple functionalities in the resistance studies of materials under use. Where external values can be quantified and taken into account prior to the development of various elements.
In general, Green's theorem facilitates the understanding and definition of the areas where vector functions are defined with respect to a region along a path.
History
It was published in 1828 in the work Mathematical analysis to the theories of electricity and magnetism, written by the British mathematician George Green. In it, quite decisive sections in the application of calculus in physics are explored, such as the concept of potential functions, Green's functions and the applications of his self-titled theorem.
George Green formalized his student career at age 40, being until now a completely self-taught mathematician. After studying at the University of Cambridge, he continued his research, making contributions in acoustics, optics and hydrodynamics that are still valid today.
Relationship with other theorems
Green's theorem is a special case, and it arises from 2 other very important theorems in the field of calculus. These are the Kelvin-Stokes theorem and the divergence or Gauss Ostrogradski theorem.
Starting from either of the two theorems, one can arrive at Green's theorem. Certain definitions and propositions are necessary to develop such proofs.
Exercises
- The following exercise shows how to transform a line integral into a double integral with respect to a region R.
The original expression is the following:
From where the corresponding functions af and g are taken
f (x, y) = x 3 g (x, y) = yx
df / dy = 0 dg / dx = y
There is no single way to define the limits of integration when applying Green's theorem. But there are ways where the integrals after being defined can be simpler. So the optimization of the integration limits deserves attention.
Where when solving the integrals we obtain:
This value corresponds in cubic units to the region below the vector function and over the triangular region defined by C.
In the case of the line integral without performing Green's method, it would have been necessary to parameterize the functions in each section of the region. That is, perform 3 parameterized integrals for the resolution. This is sufficient evidence of the efficiency that Robert Green brought with his theorem to the calculus.
References
- Introduction to Continuum Mechanics. W Michael Lai, David H. Rubin, Erhard Krempl, David Rubin Butterworth-Heinemann, Jul 23. 2009
- Multivariate Calculus. James Stewart. Cengage Learning, Mar 22 2011
- An Informal History of Green's Theorem and Associated Ideas. James Joseph Cross. Department of Mathematics, University of Melbourne, 1975
- Heat Conduction Using Greens Functions. Kevin D. Cole, James V. Beck, A. Haji-Sheikh, Bahman Litkouhi. Taylor & Francis, Jul 16 2010
- Application of Green's Theorem to the Extremization of Linear Integrals. Defense Technical Information Center, 1961