CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND x2+y2+z2=25 Q6. Evaluate over the hemisphere defined by with , F  d S z 0 S x where    F yˆi 2zˆj ykˆ GREEN’S THEOREM: Let C be a simple closed curve in the xy- plane such that a line parallel to either axis C M, N cuts in at most two points. Let and be continuous functions and (x,y) C. C. of inside and on let R be the region inside dxdy R C where is traversed in the positive (counter clockwise) sense. Y PROOF Let the equations of the curves AEB and AFB (see y=Y1(x) y=Y2(x) adjoining figure) be and respectively. If F f B A E we have x, dx b x Y2 b x Y2 X O yY1x b b a a a b C Then
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS Q6. Evaluate F  d S over the hemisphere defined by x2+y2+z2=25 with z 0 , S where F  x  yˆi 2zˆj  ykˆ GREEN’S THEOREM: Let C be a simple closed curve in the xy- plane such that a line parallel to either axis cuts C in at most two points. Let M, N and and be continuous functions of (x,y) inside and on C. let R be the region inside C. R dxdy where C is traversed in the positive (counter clockwise) sense. PROOF Let the equations of the curves AEB and AFB (see adjoining figure) be y=Y1(x) and y=Y2(x) respectively. If f Y F A B we have E Y2 x b x, Y2 x yY1x dx O b X b b a a a b C Then