CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND 2 xˆ Then a unit normal is  n  .   xˆi yˆj 2 2x2 dS3 F dS3  y3  nˆ S3 S3 Now since the radius of the cylinder 2; in cylindrical polar coordinates; we have x  2 cos , y  2 sin , z  z so that dS3  2ddz 2 3 2 3 2 S3 Then the surface integral . Hence verified.     0 36 48 84 S NB: Evaluation of the surface integral over could also have been done by projection of S3 xz yz on the or coordinate planes. QUESTIONS: Q1. Verify the Gauss divergence theorem for the vector field 2 F  xiˆ   z ˆj 2kˆ taken over the region bounded by the planes z=0, z=4, x=0, y=0 and the surface x2+y2=4 in the first octant. Ans. 20 S Q2. Evaluate where and is the surface of the cube F dS F  4 y   nˆ xzˆi ˆj yzkˆ 2 S x=0, x=1, y=0, y=1, z=0, z=1. bounded by Ans. 3 2 Q3. Evaluate , where S is a closed surface. r  nˆ dS S Ans. 3V dV  Q4. Prove that .  d S V S denote the position vector of any point (x,y,z) Q5. Let S be a closed surface and let r measured from an origin O. prove that: 74 S 3 0 if 0 lies outside S . if 0 lies inside S .
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS Then a unit normal is n    2 . xˆi  yˆj 2 F  nˆ dS3  2x2 y3 dS3 S3 S3 Now since the radius of the cylinder 2; in cylindrical polar coordinates; we have x  2 cos , y  2 sin , z  z so that dS3  2ddz 2 3 2 3 2 S3 Then the surface integral  0  36  48  84 . Hence verified. NB: Evaluation of the surface integral over S could also have been done by projection of S3 on the xz or yz coordinate planes. QUESTIONS: Q1. Verify the Gauss divergence theorem for the vector field F  xiˆ  2 ˆj  z 2kˆ taken over the region bounded by the planes z=0, z=4, x=0, y=0 and the surface x2+y2=4 in the first octant. Ans. 20 Q2. Evaluate F  nˆ dS where F  4 xzˆi y 2 ˆj  yzkˆ and S is the surface of the cube S bounded by x=0, x=1, y=0, y=1, z=0, z=1. Ans. 3 2 Q3. Evaluate r  nˆ dS , where S is a closed surface. S Ans. 3V Q4. Prove that dV   d S . V S Q5. Let S be a closed surface and let r denote the position vector of any point (x,y,z) measured from an origin O. prove that: S 3 0 if 0 lies outside S . if 0 lies inside S . 74 xˆ