∞Calculus IV Unit 15 Review

Spherical coordinates describe points in 3D space using a radius and two angles instead of three perpendicular distances. This system simplifies problems with spherical symmetry, like computing the volume of a sphere or integrating a radial field, because the geometry of the region matches the coordinate system naturally.

This section covers the three spherical coordinates, how to convert between spherical and Cartesian systems, and how the volume element $dV = \rho^2 \sin\theta\, d\rho\, d\theta\, d\phi$ arises from the Jacobian determinant.

Spherical Coordinate System

Coordinate System Components

A point in spherical coordinates is specified by three values: $(\rho, \theta, \phi)$ .

$\rho$ (rho) is the radial distance from the origin to the point. It's always non-negative: $\rho \geq 0$ .
$\theta$ (theta) is the polar angle, measured down from the positive $z$ $z$ -axis to the line connecting the origin to the point. Its range is $0 \leq \theta \leq \pi$ $0 \leq θ \leq π$ .
- $\theta = 0$ : positive $z$ -axis
- $\theta = \pi/2$ : the $xy$ -plane
- $\theta = \pi$ : negative $z$ -axis
$\phi$ (phi) is the azimuthal angle, measured in the $xy$ $x y$ -plane from the positive $x$ $x$ -axis to the projection of the point. Its range is $0 \leq \phi < 2\pi$ $0 \leq ϕ < 2 π$ .
- $\phi = 0$ : positive $x$ -axis
- $\phi = \pi/2$ : positive $y$ -axis
- $\phi = \pi$ : negative $x$ -axis
- $\phi = 3\pi/2$ : negative $y$ -axis

Convention warning: Some textbooks (especially in physics) swap the roles of $\theta$ and $\phi$ . In this course, $\theta$ is the polar angle from the $z$ -axis and $\phi$ is the azimuthal angle in the $xy$ -plane. Always check which convention your text uses.

Coordinate Surfaces

Each coordinate, when held constant, traces out a characteristic surface. These are useful for visualizing integration regions.

Constant $\rho = \rho_0$ : a sphere of radius $\rho_0$ centered at the origin.
Constant $\theta = \theta_0$ : a circular cone with its vertex at the origin and its axis along the $z$ -axis. The half-angle of the cone is $\theta_0$ . (The special case $\theta_0 = \pi/2$ gives the $xy$ -plane.)
Constant $\phi = \phi_0$ : a vertical half-plane that contains the $z$ -axis and makes angle $\phi_0$ with the $xz$ -plane.

Coordinate Transformations

Coordinate System Components, Cylindrical and Spherical Coordinates · Calculus

Spherical to Cartesian

Given $(\rho, \theta, \phi)$ , the Cartesian coordinates are:

$x = \rho \sin\theta \cos\phi$

$y = \rho \sin\theta \sin\phi$

$z = \rho \cos\theta$

These follow directly from trigonometry. The factor $\rho\sin\theta$ is the distance from the point's projection in the $xy$ -plane to the $z$ -axis (i.e., the cylindrical radius $r$ ). Then $x$ and $y$ come from resolving that projection using $\phi$ .

Cartesian to Spherical

Given $(x, y, z)$ , the spherical coordinates are:

$\rho = \sqrt{x^2 + y^2 + z^2}$

$\theta = \arccos\!\left(\frac{z}{\rho}\right), \quad \rho \neq 0$

$\phi = \text{atan2}(y, x)$

The formula for $\phi$ requires care with quadrants. In practice, use the two-argument arctangent (atan2) or apply these cases:

If $x > 0$ : $\phi = \arctan(y/x)$
If $x < 0$ and $y \geq 0$ : $\phi = \arctan(y/x) + \pi$
If $x < 0$ and $y < 0$ : $\phi = \arctan(y/x) + \pi$
If $x = 0$ and $y > 0$ : $\phi = \pi/2$
If $x = 0$ and $y < 0$ : $\phi = 3\pi/2$
If $x = 0$ and $y = 0$ : $\phi$ is undefined (the point is on the $z$ -axis)

Note: when $x < 0$ and $y < 0$ , you add $\pi$ (not subtract) to land in the correct range $[0, 2\pi)$ .

Jacobian Determinant

When you change variables in a triple integral, you need the Jacobian determinant to account for how volumes distort under the transformation. For the spherical-to-Cartesian map, the Jacobian matrix is:

\frac{\partial x}{\partial \rho} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial \phi} \$4pt] \frac{\partial y}{\partial \rho} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial \phi} \$4pt] \frac{\partial z}{\partial \rho} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial \phi} \end{pmatrix} = \begin{pmatrix} \sin\theta\cos\phi & \rho\cos\theta\cos\phi & -\rho\sin\theta\sin\phi \\ \sin\theta\sin\phi & \rho\cos\theta\sin\phi & \rho\sin\theta\cos\phi \\ \cos\theta & -\rho\sin\theta & 0 \end{pmatrix}$$ Computing the determinant (expand along the third row or use cofactors) gives: $$|J| = \rho^2 \sin\theta$$ The factor $$\rho^2$$ reflects how volume scales with distance from the origin (think of how spherical shells get larger). The $$\sin\theta$$ factor accounts for the "compression" of volume near the poles ($$\theta = 0$$ and $$\theta = \pi$$), similar to how lines of longitude converge at the Earth's poles. ###### ![fiveable_print_image_2](https://fiveable.me) ## Integration in Spherical Coordinates ### Volume Element The volume element in spherical coordinates combines the Jacobian with the three differentials: $$dV = \rho^2 \sin\theta\, d\rho\, d\theta\, d\phi$$ You can also understand this geometrically. A small "spherical box" at position $$(\rho, \theta, \phi)$$ has: - Radial thickness: $$d\rho$$ - Arc length in the $$\theta$$-direction: $$\rho\, d\theta$$ - Arc length in the $$\phi$$-direction: $$\rho\sin\theta\, d\phi$$ Multiplying these three lengths gives the volume of the box: $$\rho^2 \sin\theta\, d\rho\, d\theta\, d\phi$$. ### Setting Up Triple Integrals A triple integral over a region $$E$$ in spherical coordinates takes the form: $$\iiint_E f(\rho, \theta, \phi)\, \rho^2 \sin\theta\, d\rho\, d\theta\, d\phi$$ The typical integration order is $$d\rho$$ first (innermost), then $$d\theta$$, then $$d\phi$$ (outermost). For a full sphere of radius $$R$$, the limits are: - $$0 \leq \rho \leq R$$ - $$0 \leq \theta \leq \pi$$ - $$0 \leq \phi \leq 2\pi$$ For regions that aren't full spheres (cones, spherical caps, wedges), the limits on $$\theta$$ and $$\phi$$ will be restricted accordingly, and $$\rho$$ limits may depend on $$\theta$$ or $$\phi$$. Identifying the correct limits from the geometry of the region is usually the hardest part of setting up these integrals.

∞Calculus IV Unit 15 Review