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 div𝑓 and curl𝑓, where 𝑓(π‘Ÿ,πœƒ,𝑧)=π‘Ÿπ‘’+π‘§πœƒπ‘’+π‘Ÿπ‘§π‘’οŽοΌο™sin.

  • Adivcos𝑓=2π‘Ÿ+π‘Ÿ+π‘§πœƒπ‘Ÿ, curlcossin𝑓=(π‘Ÿβˆ’π‘§πœƒ)π‘’βˆ’π‘§π‘’+π‘§πœƒπ‘Ÿπ‘’οŽοΌο™
  • Bdivcos𝑓=2π‘Ÿ+π‘Ÿ+2π‘§πœƒπ‘Ÿ, curlcossin𝑓=(π‘Ÿβˆ’π‘§πœƒ)π‘’βˆ’π‘§π‘’+π‘§πœƒπ‘Ÿπ‘’οŽοΌο™
  • Cdivcos𝑓=2π‘Ÿ+π‘Ÿ+π‘§πœƒπ‘Ÿ, curlcossin𝑓=(π‘Ÿ+2π‘§πœƒ)π‘’βˆ’π‘§π‘’+π‘§πœƒπ‘Ÿπ‘’οŽοΌο™
  • Ddivcos𝑓=2+π‘Ÿβˆ’π‘§πœƒπ‘Ÿ, curlcossin𝑓=(1+π‘§πœƒ)π‘’βˆ’π‘§π‘’βˆ’πœƒπ‘Ÿπ‘’οŽοΌο™
  • Edivcos𝑓=2+π‘Ÿ+π‘§πœƒπ‘Ÿ, curlsinsin𝑓=(βˆ’πœƒ)π‘’βˆ’π‘§π‘’+π‘§πœƒπ‘Ÿπ‘’οŽοΌο™

Q2:

Use spherical coordinates to find div𝑓 and curl𝑓, where 𝑓(𝜌,πœƒ,πœ™)=𝑒+πœŒπœƒπ‘’+πœŒπ‘’οŽ…οΌοŽ’cos.

  • Adivcossin𝑓=2𝜌+2πœƒπœƒ, curlcotcos𝑓=πœƒπ‘’βˆ’2𝑒+2πœƒπ‘’οŽ…οΌοŽ’
  • Bdivcossin𝑓=2πœŒβˆ’2πœƒπœƒ, curlcotcos𝑓=πœƒπ‘’βˆ’2π‘’βˆ’2πœƒπ‘’οŽ…οΌοŽ’
  • Cdivcossin𝑓=2πœŒβˆ’2πœƒπœƒ, curlcotcos𝑓=πœƒπ‘’βˆ’2𝑒+2πœƒπ‘’οŽ…οΌοŽ’
  • Ddivsincoscot𝑓=2𝜌+πœƒπœ™+πœ™, curlcotcoscos𝑓=πœƒπœƒπ‘’+2𝑒+2πœƒπ‘’οŽ…οΌοŽ’
  • Edivcossin𝑓=2𝜌+2πœƒπœƒ, curlcotcos𝑓=πœƒπ‘’βˆ’2π‘’βˆ’2πœƒπ‘’οŽ…οΌοŽ’

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