THE GREENS THEOREM

When I was a student at a technical school I had real problem with this theorem and thought to myself why not try to simply this theorem to any present day student who would be having difficulty at understanding this theorem.

I have seen many people who forcibly digest these theorems with no understanding what so ever of what they are doing. Mathematics and its many variants have been confused profusely by humanity in some form or another so much so that many practitioners fail to assimilate or atleast ponder to give many things a thought.

They say Greens theorem is another version of the stokes theorem about which I have no idea but will dwell upon it some time in the future.

At the center of the greens theorem is the cartesian co-ordinates which can be treated both as a vector or scalar co-ordinate. Note that scalars do not have direction as against thevectors. When some one uses the cartesian co=ordinates in a polar form it becomes vector co-ordinates and is defined by r and theta..r signofying the magnitude and theta signifying the direction. Whenever we define and treat the cartesian co-ordinates by x and y they are scalar co-ordinates. Hence we have both scalar and vector algebra arising from the cartesian plane.

What are the basic components of the Greens Theorem;

1. A closed curve chosen to be traversed in the counter clockwise direction also called positive direction.

2. Another function defined by X and Y components called F which is integrated upon the above curve which can be smooth or piece-wise smooth.( A circle is smooth whereas a triangle is piece wise smooth).

3. Greens theorem like many simple theorems says that the line integral over the curve is equal to a partial double integral over the plane.

The theorem goes something like this;

Closed Intregral of f.dx + g.dy along the curve mentioned

equals

Double Integral of the function resulting from the subtraction of PD of g with respect to x and PD of f with respect to y.

(the above will be mathematically described by any text)

PD means partial derivative

f is the x component of the plane F which will be a function in x and y

g is the y component of the plane F which will be a function of x and y

Example : Let F be defined as ( -y x) and let the curve in question be a circle x^2 + y^2 = 4 which would be a circle centered at the origin with radius 2.

Note that at the end of the day "What does Green's theorem do ?" that is very simple. A line integral of a function over a closed loop is calculated easily using a partial differential equivalent.

In the above problem suggested f is -y and g is x and please compute the partial integrals and you get 8 phi which is twice the area of the circle.

IF YOU ARE A STUDENT OF MATHEMATICS YOU MIGHT HAVE TO GO MORE INTO THE DETAIL OF THIS THEOREM. IF YOU ARE A STUDENT OF ENGINEERING GET THE ABOVE IDEA WHICH I HAVE GIVEN AND ALSO LEARN THE MATHEMATICAL PROOF AND SOLVE SOME SIMPLE PROBLEMS. THIS AREA IS SOMETIMES CALLED COMPLEX ALGEBRA NOT TO BE CONFUSED AND FRIGHTENED ABOUT..THIS IS JUST AN ALGEBRA OF TWO FUNCTIONS ON THE CARTESIAN PLANE.

GOOD LUCK