Green Theorem

This presentation contains the introduction, proof, and example problem of the green theorem

1. GREEN THEOREM (TRANSFORMATION OF DOUBLE INTEGRALS TO LINE INTEGRALS) COMPLEX VARIABLE AND LAPLACE TRANSFORM PRESENTED BY: Sarwan Ahmed Ursani (18CH101) ASSIGNED BY: Sir Ayaz Siyal Mehran University Of Engineering & Technology

2. INTRODUCTION This law was proposed by George Green in 1828 A.D. and is named after him. In mathematics, Green's theorem gives the relationship between a  line integral  around, a simple closed curve  C  and a  double integral  over the plane region  D  bounded by  C . The Green theorem is used to transform double integrals over a plane region into line integral over the boundary of the region. Identities derived from Green's theorem play a key role in  reciprocity in electromagnetism It makes the contour integral easier. NOTE: In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.

3. STATEMENT OF GREEN THEOREM Let R be a closed bounded region in the xy - plane, whose boundary C consists of finitely many smooth curves. Let P( x,y ) and Q( x,y ) be the continuous function having continuous partial derivative. In region R and on its boundary C, Green theorem states that =    

4. PROOF OF GREEN THEOREM Consider a simple closed curve C bounding the region R as show in figure. Let us divide the curve C into two parts ACB and ADB and let us suppose that the equations of these curves be y= f 1 (x) and y = f 2 (x) respectively. Now R is the region bounded by C, hence  

5. = - Hence Now consider the curve CAD and CBD. Let their equations be y = g 1 (x) and y = g 2 (x) this implies that x = g 1 -1 (y) = G 1 (y) and x = g 2 -2 (y) = G 2 (y)  

6. = - Hence Subtracting Eq ( i ) From Eq (ii) + = OR = - This Proves Green Theorem  

7. Exercise Problem Q: Verify the Green’s theorem for where C is simple closed curve enclosed by the parabolas y = x 2 and x = y 2 .   First we will solve this problem by contour integration. We have two equation, y = x 2 …( i ) and x = y 2 …(ii) x 4 = x x 4 – x = 0 x(x 3 -1) = 0 We get x = 0, x = 1 Now when x =0 , y = 0 & x= 1, y = 1. We can say that the point of intersection between the two given curve are (0,0) and (1,1).

8. Along C 1 y = x 2 so that dy = 2x and x varies from 0 to 1  

9. Along C 2 x = y 2 so that dx = 2y and y varies from 1 to 0 Now, C = C 1 + C2  

10. Now, using Green Theorem We can solve this problem easily Here P = & Q = A/C to Green Theorem = - &    

11. This is proves the green theorem for the given contour integral over the region R.  

12. THANKYOU