CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND Let x=cos t, y = sin t,  0 , 0 2 be parametric equations of C. Then z t sin   t)dt 2 2 C   o (2 cos sin t i Also,   F k  kˆ j 2x y yz y2 z  Then dxdy s s R since  and R is the projection of S on the xy- plane. This last integral nˆ kˆds dxdy  equals 0 1 1 0 0 1 And the Stokes’ theorem is verified. Example 5.27 9 z=4 A surface S consists of that part of the cylinder   between  0 and x y z 2 2 for y≥0 and the two semicircles of radius 3 in the planes 0 and  4 . If  z z    S The surface S consists of three sections a. the curved surface of the cylinder b. the top and bottom semicircles. We therefore evaluate S   81
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS Let x=cos t, y = sin t, z  0 , 0 t 2 be parametric equations of C. Then   C o 2 2 (2 cos t sin sin t)dt   Also,  F  i j k  kˆ 2x y yz y2 z Then  s s dxdy R since nˆ  kˆds  dxdy and R is the projection of S on the xy- plane. This last integral equals 1 1 0 0 1 0 And the Stokes’ theorem is verified. Example 5.27 A surface S consists of that part of the cylinder x 2  y 2  9 between z  0 and z=4 for y≥0 and the two semicircles of radius 3 in the planes z  0 and z  4 . If  S   The surface S consists of three sections a. the curved surface of the cylinder b. the top and bottom semicircles. We therefore evaluate S   81