Section 12.13

Spherical Coordinates

A simplified way to describe spheres, cones, and point-symmetric systems.

Introduction

Spherical Coordinates use distance from origin and two angles.
- (rho): Distance from the origin. .
- (theta): Same as cylindrical/polar. Angle in xy-plane.
- (phi): Angle "down" from the positive z-axis. .

Interactive: The Spherical Frame

Rho (ρ) is the distance from center.
Phi (φ) is the angle down from the North Pole (Z-axis).
Theta (θ) is the angle around the equator (XY-plane).

Key Formulas

Spherical to Cartesian

Cartesian to Spherical

Common Surfaces

: Sphere.
: Cone (Half-cone).
: Vertical Half-Plane.

Worked Examples

Example 1: Identifying Surfaces (Level 1)

Identify the surface given by .

Definition of is distance from origin.
Points at constant distance 5 from origin form a Sphere of radius 5.

Identify .
Constant angle from Z-axis. This forms a Cone opening upwards.

Example 2: Conversion (Level 2)

Convert Cartesian to Spherical.

1. Find Rho

2. Find Phi

Use :.
Since , .

3. Find Theta

Use . Point is in Q2.
. Angle in Q2 is .

Result: .

Example 3: Equation Conversion (Level 3)

Convert to spherical coordinates.

LHS: .

RHS: .

Equation: .

Simplify (assuming ):
.

(This represents a sphere shifted up the z-axis).