Cylindrical coordinates offer a fresh perspective on 3D space, using distance from the z-axis, angle, and height. This system simplifies calculations for objects with circular symmetry, like cylinders and cones, making it a powerful tool for certain types of problems.

Transforming between Cartesian and cylindrical coordinates involves key formulas and the Jacobian determinant. Understanding these transformations is crucial for solving triple integrals in cylindrical coordinates, which we'll explore further in this unit.

Cylindrical Coordinate System

Defining Cylindrical Coordinates

• Cylindrical coordinate system represents points in 3D space using $(\rho, \phi, z)$
• $\rho$ represents the radial distance from the z-axis to the point in the xy-plane
  • Always non-negative ($\rho \geq 0$)
  • Calculated using the formula $\rho = \sqrt{x^2 + y^2}$
• $\phi$ represents the azimuthal angle in the xy-plane from the positive x-axis
  • Measured counterclockwise
  • Range is $0 \leq \phi < 2\pi$
  • Calculated using the formula $\phi = \tan^{-1}(\frac{y}{x})$
• $z$ represents the height or distance along the z-axis
  • Can be positive, negative, or zero
  • Corresponds directly to the z-coordinate in the Cartesian system

Relationship to Cartesian Coordinates

• Cylindrical coordinates can be converted to Cartesian coordinates using the following formulas:
  • $x = \rho \cos(\phi)$
  • $y = \rho \sin(\phi)$
  • $z = z$
• Cartesian coordinates can be converted to cylindrical coordinates using the following formulas:
  • $\rho = \sqrt{x^2 + y^2}$
  • $\phi = \tan^{-1}(\frac{y}{x})$
  • $z = z$
• The z-coordinate remains the same in both coordinate systems

Coordinate Transformation

Cartesian to Cylindrical Transformation

• To transform an integral from Cartesian to cylindrical coordinates, substitute the following:
  • $x = \rho \cos(\phi)$
  • $y = \rho \sin(\phi)$
  • $z = z$
• The limits of integration must also be transformed to correspond with the cylindrical coordinate system
  • x-limits transform to $\rho$-limits
  • y-limits transform to $\phi$-limits
  • z-limits remain the same
• Example: $\int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \int_{0}^{1} dzdydx$ transforms to $\int_{0}^{2\pi} \int_{0}^{1} \int_{0}^{1} \rho dzd\rho d\phi$

Jacobian Determinant and Volume Element

• When transforming integrals, the Jacobian determinant is introduced to account for the change in volume elements
• The Jacobian determinant for the transformation from Cartesian to cylindrical coordinates is $\rho$
• The volume element in cylindrical coordinates is $dV = \rho d\rho d\phi dz$
  • $\rho$ comes from the Jacobian determinant
  • $d\rho$, $d\phi$, and $dz$ represent the infinitesimal changes in each coordinate
• Example: $\iiint_D f(x,y,z) dxdydz = \iiint_D f(\rho \cos(\phi), \rho \sin(\phi), z) \rho d\rho d\phi dz$
  • $D$ is the domain of integration in cylindrical coordinates
  • $f(x,y,z)$ is transformed to $f(\rho \cos(\phi), \rho \sin(\phi), z)$
  • $dxdydz$ is replaced by the volume element $\rho d\rho d\phi dz$