CHAPTER 5 VECTOR INTEGRATION INTEGRAL THEOREMS AND In case of a smooth surface S given by an equation Z=f(x,y) defined over a region Rxy in the xy- plane, we write F(x,y,z)=f(x,y)-z=0 S and consider as the level surface F(x,y,z)=0 of the function F. Taking the normal to be to Rxy we have  nˆ kˆ  f  f kˆ  ˆj xiˆ y 2 2 1    nˆ kˆ  k S  2 2 dA  Rxy  nˆ Rxy Similarly, formulations are easily given for the area of smooth surface x=f(y,z) over a region Ryz and y=f(x,z) over region Rxz. Example 5.9 Find the area of the upper cap cut from the sphere by the cylinder    2 x y z 2 2 2   1. x y 2 2 Z    x 2 y 2 z 2 2   2 y 2 1 Y 1 1 X x, The cap part of the surface , projects onto the disc     2 F y, z x y z 2 2 2 the xy- plane. At any point (x,y,z) in the space we have : 1in   R y 2 2    F y z 2 2 2  2xiˆ  2  2zkˆ yˆj 2  x  y  z 2 2 2 On the level surface 2 F  x  y  z  2 2 2 2 2  63
CHAPTER 5 VECTOR INTEGRATION AND INTEGRAL THEOREMS In case of a smooth surface S given by an equation Z=f(x,y) defined over a region Rxy in the xy- plane, we write F(x,y,z)=f(x,y)-z=0 and consider S as the level surface F(x,y,z)=0 of the function F. Taking the normal to be nˆ  kˆ to Rxy we have  f xiˆ  f y ˆj kˆ   nˆ  kˆ  k  1 2 2 S  Rxy  nˆ dA  Rxy 2 2 Similarly, formulations are easily given for the area of smooth surface x=f(y,z) over a region Ryz and y=f(x,z) over region Rxz. Example 5.9 Find the area of the upper cap cut from the sphere x 2  y 2  z 2  2 by the cylinder x 2  y 2  1. Z x 2  y 2  z 2  2 2  y 2  1 1 1 Y X The cap part of the surface F x, y, z  x 2  y 2  z 2  2 , projects onto the disc R : 2  y 2  1in the xy- plane. At any point (x,y,z) in the space we have F  2  y 2  z 2  2xiˆ  2 yˆj  2zkˆ  2 x 2  y 2  z 2 On the level surface F  x 2  y 2  z 2  2  2 2 63