Paul's Online Notes- Spherical Coordinates Tutorial

What Is Paul's Online Notes?

Paul's Online Notes is a free math resource created by Paul Dawkins, a professor at Lamar University. The site covers calculus I, II, and III, along with differential equations and linear algebra. It's been around since the early 2000s and remains one of the most-used supplementary math resources on the web.

The site is bare-bones. No videos, no animations, just clean explanations and worked examples. Students either love it or find it too dry. But if you need to actually understand the material, the clarity is hard to beat.

What Are Spherical Coordinates?

Spherical coordinates describe points in 3D space using three values: ρ (rho), θ (theta), and φ (phi).

Think of it like this: you're standing at the origin. ρ tells you how far to walk. θ tells you which direction to turn in the horizontal plane. φ tells you how much to tilt up or down from the horizontal.

• ρ = distance from origin (always ≥ 0)
• θ = angle in the xy-plane from the positive x-axis (same as cylindrical coordinates)
• φ = angle from the positive z-axis (0 ≤ φ ≤ π)

The key difference from cylindrical coordinates: φ measures from the z-axis, not from the xy-plane. This trips people up constantly.

Spherical to Cartesian Conversion

You'll need these formulas constantly:

• x = ρ sin φ cos θ
• y = ρ sin φ sin θ
• z = ρ cos φ

The sin φ term appears because you're projecting the radial distance onto the xy-plane before adding the x and y components.

Cartesian to Spherical Conversion

Working backwards:

• ρ = √(x² + y² + z²)
• tan θ = y/x (watch your quadrant)
• cos φ = z/ρ

When x = 0 and y = 0, you're on the z-axis. θ can be anything; pick 0. When ρ = 0, you're at the origin—θ and φ are irrelevant.

Common Surfaces in Spherical Coordinates

Equations simplify dramatically in spherical form:

• ρ = constant → sphere centered at origin
• φ = constant → cone (or half-plane when φ = 0 or π)
• θ = constant → half-plane containing the z-axis
• ρ = a cos φ → sphere of radius a/2 centered on the z-axis
• ρ = a sin φ → sphere of radius a/2 centered on the x-axis

The last two are particularly useful. You can replace messy Cartesian equations with these simple forms.

Triple Integrals in Spherical Coordinates

This is where spherical coordinates actually matter. The volume element becomes:

dV = ρ² sin φ dρ dφ dθ

When your region or integrand involves spheres or cones, switch to spherical. The integral becomes orders of magnitude simpler.

Integration bounds typically run:

• θ: 0 to 2π (full rotation)
• φ: 0 to π (from positive z-axis to negative z-axis)
• ρ: 0 to some function of θ and φ, or a constant

How to Use Paul's Online Notes for Spherical Coordinates

Here's the practical approach:

1. Read the main explanation page. It covers the coordinate system, conversions, and common surfaces. Maybe 20-30 minutes if you're working through examples.
2. Skim the examples section. Paul works through Cartesian-to-spherical conversions, surface identification, and triple integral setup.
3. Try the practice problems. The site doesn't auto-grade, but the solutions are fully worked.

The site organizes content by course. Look for "Calculus III" then "Triple Integrals" or "Cylindrical and Spherical Coordinates."

Spherical Coordinates vs. Cylindrical: When to Use Which

There's no strict rule. Try one system, see if the bounds simplify, switch if they don't.

Common Mistakes

• Confusing φ with θ. φ is from the z-axis. θ is from the x-axis in the xy-plane.
• Forgetting sin φ in the volume element for triple integrals.
• Using the wrong range for φ. It goes 0 to π, not 0 to 2π.
• Not checking quadrants when computing θ from tan θ = y/x.

Bottom Line

Paul's Online Notes gives you exactly what you need: definitions, formulas, and worked examples without the filler. For spherical coordinates, work through the conversion examples and the triple integral examples. That's it. The site won't hold your hand, but it'll get you through the material if you're willing to actually work the problems.