Worksheet: Divergence and Curl in Cylindrical and Spherical Coordinates

In this worksheet, we will practice finding the divergence and the curl of a vector field in cylindrical and spherical coordinates.

Q1:

Use cylindrical coordinates to find d i v 𝑓 and c u r l 𝑓 , where 𝑓 ( 𝑟 , 𝜃 , 𝑧 ) = 𝑟 𝑒 + 𝑧 𝜃 𝑒 + 𝑟 𝑧 𝑒 s i n .

  • A d i v c o s 𝑓 = 2 𝑟 + 𝑟 + 𝑧 𝜃 𝑟 , c u r l c o s s i n 𝑓 = ( 𝑟 + 2 𝑧 𝜃 ) 𝑒 𝑧 𝑒 + 𝑧 𝜃 𝑟 𝑒
  • B d i v c o s 𝑓 = 2 + 𝑟 𝑧 𝜃 𝑟 , c u r l c o s s i n 𝑓 = ( 1 + 𝑧 𝜃 ) 𝑒 𝑧 𝑒 𝜃 𝑟 𝑒
  • C d i v c o s 𝑓 = 2 + 𝑟 + 𝑧 𝜃 𝑟 , c u r l s i n s i n 𝑓 = ( 𝜃 ) 𝑒 𝑧 𝑒 + 𝑧 𝜃 𝑟 𝑒
  • D d i v c o s 𝑓 = 2 𝑟 + 𝑟 + 𝑧 𝜃 𝑟 , c u r l c o s s i n 𝑓 = ( 𝑟 𝑧 𝜃 ) 𝑒 𝑧 𝑒 + 𝑧 𝜃 𝑟 𝑒
  • E d i v c o s 𝑓 = 2 𝑟 + 𝑟 + 2 𝑧 𝜃 𝑟 , c u r l c o s s i n 𝑓 = ( 𝑟 𝑧 𝜃 ) 𝑒 𝑧 𝑒 + 𝑧 𝜃 𝑟 𝑒

Q2:

Use spherical coordinates to find d i v 𝑓 and c u r l 𝑓 , where 𝑓 ( 𝜌 , 𝜃 , 𝜙 ) = 𝑒 + 𝜌 𝜃 𝑒 + 𝜌 𝑒 c o s .

  • A d i v s i n c o s c o t 𝑓 = 2 𝜌 + 𝜃 𝜙 + 𝜙 , c u r l c o t c o s c o s 𝑓 = 𝜃 𝜃 𝑒 + 2 𝑒 + 2 𝜃 𝑒
  • B d i v c o s s i n 𝑓 = 2 𝜌 + 2 𝜃 𝜃 , c u r l c o t c o s 𝑓 = 𝜃 𝑒 2 𝑒 + 2 𝜃 𝑒
  • C d i v c o s s i n 𝑓 = 2 𝜌 2 𝜃 𝜃 , c u r l c o t c o s 𝑓 = 𝜃 𝑒 2 𝑒 + 2 𝜃 𝑒
  • D d i v c o s s i n 𝑓 = 2 𝜌 + 2 𝜃 𝜃 , c u r l c o t c o s 𝑓 = 𝜃 𝑒 2 𝑒 2 𝜃 𝑒
  • E d i v c o s s i n 𝑓 = 2 𝜌 2 𝜃 𝜃 , c u r l c o t c o s 𝑓 = 𝜃 𝑒 2 𝑒 2 𝜃 𝑒

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