Worksheet: Jacobians and Change of Variables in Multiple Integrals

In this worksheet, we will practice making change of variables in multiple integrals and finding the Jacobian of the function of transformation of variables.

Q1:

Give an expression for the Jacobian 𝜕 ( 𝑥 , 𝑦 ) 𝜕 ( 𝑟 , 𝜃 ) for the change of variables from polar to Cartesian coordinates on .

  • A c o s s i n 𝜃 𝜃
  • B 𝑟
  • C 𝑟 ( 2 𝜃 ) c o s
  • D 𝑟
  • E 𝑟 ( 𝜃 𝜃 ) c o s s i n

Q2:

Give an expression for the Jacobian 𝜕 ( 𝑟 , 𝜃 ) 𝜕 ( 𝑥 , 𝑦 ) for the change of variables from Cartesian to polar coordinates on .

  • A 𝑥 + 𝑦
  • B 1 𝑥 𝑦
  • C 𝑥 + 𝑦
  • D 1 𝑥 + 𝑦
  • E 1 𝑥 + 𝑦

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