Coordinate systems for integration

Some common change in coordinate systems to evaluate integrals better and their corresponding scaling factor to convert between systems using the Jacobian:

In 2D #

• Cartesian: dA=dxdy
• Polar: dA=r drdθ
  • x=$r\cos\theta$
  • y=$r\sin\theta$

In 3D #

• Cartesian: dV=dxdydz
• Cylindrical: dV=r drdθdz
  • x=$r\cos\theta$
  • y=$r\sin\theta$
  • z=z
• Spherical: dV=$r^2 \sin(\theta)drd\theta d\phi$
  • x=$r\sin \theta \cos\phi$
  • y=$r\sin \theta \sin\phi$
  • z=$r \cos \theta$