CHAPTER 6 CURVILINEAR COORDINATES dV b) Element of volume (hU1 ) (hU )   dV dU1eˆ1 dU hU3 dU  2 2eˆ2 3eˆ3  hU1 hU2 hU3 dU1dU dU 2 3 ) dU1dU 2dU 3 y z)  dU1dU JdU1dU dU 2dU 3 2 3 )  U U 2 3 where J is called the Jacobian of the transformation and is given by  xU2 yU2 zU2 y z) xU3 yU3 zU3 )  U U J 2 3 yU1 zU1 xU1 etc. where xU1  dA c) Element of Area The area element is given by   from fig. A   dA1 hU hU3 dU 2dU 2eˆ2 3eˆ3  hU2 hU3 eˆ2  eˆ3 dU dU  hU2 hU3 dU 2 3 2dU 3 since    1 eˆ2 eˆ3 eˆ1 dU1dU 2 and  similarly dA2 hU1 hU3 dU1dU 3  dA3 Example 6.5 hU1 hU2 Evaluate   )dxdydz where V is a sphere having centre at the origin and y z 2 2 2 V radius equal to a. We may consider the portion in the first octant and in spherical coordinates we have  sindrdd dxdydz r 2
CHAPTER 6 CURVILINEAR COORDINATES b) Element of volume dV dV  (hU1 dU1eˆ1 )  (hU 2 dU 2eˆ2  hU3 dU 3eˆ3 )  hU1 hU2 hU3 dU1dU 2 dU 3 ) dU1dU 2dU 3  y z) U 2 U 3 ) dU1dU 2dU 3  JdU1dU 2 dU 3 where J is called the Jacobian of the transformation and is given by J  y z) U 2 U 3 ) xU1  xU2 xU3 yU1 yU2 yU3 zU1 zU2 zU3 where xU1  etc. c) Element of Area dA The area element is given by dA1  hU 2dU 2eˆ2  hU3 dU 3eˆ3 from  fig. A   hU2 hU3 eˆ2  eˆ3 dU 2 dU 3  hU2 hU3 dU 2dU 3 since eˆ2  eˆ3  eˆ1  1 similarly dA2  hU1 hU3 dU1dU 3 Example 6.5 and dA3  hU1 hU2 dU1dU 2 Evaluate V 2  y 2  z 2 )dxdydz where V is a sphere having centre at the origin and radius equal to a. We may consider the portion in the first octant and in spherical coordinates we have dxdydz  r 2 sindrdd