Coordinate Systems in Multivariable Calculus: Cylindrical and Spherical Coordinates

Ch. 12 - Parametric and Polar Curves

Coordinate Systems: Cylindrical and Spherical Coordinates

In multivariable calculus, different coordinate systems are used to simplify the representation and calculation of points and regions in three-dimensional space. The most common systems are rectangular (Cartesian), cylindrical, and spherical coordinates. Each system is suited for specific types of problems, especially those involving symmetry.

Cylindrical Coordinates

• Definition: Cylindrical coordinates represent a point in space by $(r, \theta, z)$, where:

  • $r$ is the radial distance from the $z$-axis (projection onto the $xy$-plane).

  • $\theta$ is the angle from the positive $x$-axis in the $xy$-plane.

  • $z$ is the height above the $xy$-plane.

• Conversion to Rectangular Coordinates:

  • $x = r \cos \theta$

  • $y = r \sin \theta$

  • $z = z$

• Limits for Integration:

  • $0 \leq r \leq R$ (where $R$ is the outer radius)

  • $0 \leq \theta \leq 2\pi$

  • $a \leq z \leq b$

• Volume Element: $dV = r\,dr\,d\theta\,dz$

• Example: To find the volume of a cylinder of radius $R$ and height $h$, integrate over $r$$from $0$$to$R$,$\theta$from $0$ to $2\pi$, and $z$ from $0$to$h$.

Spherical Coordinates

• Definition: Spherical coordinates represent a point in space by $(\rho, \theta, \phi)$, where:

  • $\rho$ is the distance from the origin to the point.

  • $\theta$ is the angle from the positive $x$-axis in the $xy$-plane (azimuthal angle).

  • $\phi$ is the angle from the positive $z$-axis (polar angle).

• Conversion to Rectangular Coordinates:

  • $x = \rho \sin \phi \cos \theta$

  • $y = \rho \sin \phi \sin \theta$

  • $z = \rho \cos \phi$

• Limits for Integration:

  • $0 \leq \rho \leq R$ (where $R$ is the outer radius)

  • $0 \leq \theta \leq 2\pi$

  • $0 \leq \phi \leq \pi$

• Volume Element: $dV = \rho^2 \sin \phi\,d\rho\,d\phi\,d\theta$

• Example: To find the volume of a sphere of radius $R$, integrate over $\rho$$from $0$R$,$\phi$from $0$ to $\pi$, and $\theta$ from $0$to $2\pi$.

Changing Between Coordinate Systems

• Rectangular to Cylindrical:

  • $r = \sqrt{x^2 + y^2}$

  • $\theta = \arctan\left(\frac{y}{x}\right)$

  • $z = z$

• Cylindrical to Rectangular:

  • $x = r \cos \theta$

  • $y = r \sin \theta$

  • $z = z$

• Rectangular to Spherical:

  • $\rho = \sqrt{x^2 + y^2 + z^2}$

  • $\theta = \arctan\left(\frac{y}{x}\right)$

  • $\phi = \arccos\left(\frac{z}{\rho}\right)$

• Spherical to Rectangular:

  • $x = \rho \sin \phi \cos \theta$

  • $y = \rho \sin \phi \sin \theta$

  • $z = \rho \cos \phi$

Integration in Cylindrical and Spherical Coordinates

• Triple Integrals in Cylindrical Coordinates:

  • $\iiint_{D} f(r,\theta,z)\, r\, dr\, d\theta\, dz$

• Triple Integrals in Spherical Coordinates:

  • $\iiint_{D} f(\rho,\theta,\phi)\, \rho^2 \sin \phi\, d\rho\, d\phi\, d\theta$

• Application: These integrals are used to compute volumes, masses, and other quantities over regions with cylindrical or spherical symmetry.

Comparison Table: Coordinate Systems

System	Variables	Volume Element	Typical Limits
Rectangular	$x, y, z$	$dx\,dy\,dz$	Depends on region
Cylindrical	$r, \theta, z$	$r\,dr\,d\theta\,dz$	$0 \leq r \leq R$, $0 \leq \theta \leq 2\pi$, $a \leq z \leq b$
Spherical	$\rho, \theta, \phi$	$\rho^2 \sin \phi\,d\rho\,d\phi\,d\theta$	$0 \leq \rho \leq R$, $0 \leq \theta \leq 2\pi$, $0 \leq \phi \leq \pi$

Additional info: These notes cover the essential formulas and conversion methods for cylindrical and spherical coordinates, which are crucial for evaluating triple integrals in multivariable calculus.