CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND S R We first define the area of by dividing the region into small rectangles . Directly above each there lies a patch of surface that we may approximate F(x,y,z)=C with a portion of of the tangent plane since may not be a plane. The vector and lie along the edges of the patch in the tangent plane. Both  u v v u and are normal to . Now is the area of the orthogonal projection of  v n the parallelogram determined by and onto a plane whose normal is . u v nˆ This translates into the statement u   .....( v n A) Now is the area . Hence (A) becomes  where  is the   cos  u v u v n angle between and .  u v nˆ  so that cos  provided that . cos  0   cos On summing; k or dA k R S We therefore define the area of to be the value of this integral whenever the F(x,y,z)=C integral exists. For any particular surface we have   cos nˆ n   1 nˆ  or  cos  nˆ  nˆ which results into dA Surface area   uˆ R provided . 0 nˆ 
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS We first define the area of S by dividing the region R into small rectangles . Directly above each there lies a patch of surface that we may approximate with a portion of of the tangent plane since F(x,y,z)=C may not be a plane. The vector u and v lie along the edges of the patch in the tangent plane. Both  v and are normal to . Now u  v n is the area of the orthogonal projection of the parallelogram determined by u and v onto a plane whose normal is nˆ . This translates into the statement u  v n  .....( A) Now u  v is the area . Hence (A) becomes u  v n cos   where  is the angle between u  v and nˆ . cos   so that  cos  provided that cos  0 . On summing; k or k R dA We therefore define the area of S to be the value of this integral whenever the integral exists. For any particular surface F(x,y,z)=C we have  nˆ  n cos  or 1 cos   nˆ  nˆ   nˆ which results into Surface area  R  uˆ dA provided  nˆ 0 .