Differential geometry is what happens when you turn the tools of calculus loose on curved shapes. How sharply does a road bend? How does a surface warp away from its tangent plane? Can a flat map of the round Earth ever be faithful? These are questions about curvature — the precise measure of how a shape refuses to be straight or flat — and calculus is exactly the instrument for measuring change.

Its reach is enormous. The same curvature that describes a bent wire and a saddle-shaped crisp becomes, in Einstein's hands, the very fabric of gravity: mass curves spacetime, and planets simply roll along the straightest available paths. Differential geometry is the shared language of general relativity, robotics, computer graphics and the geometry of data.

The big idea: curvature is intrinsic

One thread runs through everything here. At every point we attach a tangent — the best straight-line or flat-plane approximation — and then measure how fast the shape peels away from it. That rate of peeling is curvature. The stunning discovery, Gauss's Theorema Egregium, is that some curvature is intrinsic: an ant living on the surface could detect it without ever leaving, which is exactly why no flat map of the globe can preserve all distances.

The shape of the journey

This course climbs in three stages, from one dimension up to many.

• Stage A — Curves. One-dimensional shapes in space: how to parametrize them, measure arc length, and capture their bending with the Frenet frame.
• Stage B — Surfaces. Two-dimensional shapes: the fundamental forms, Gaussian curvature, Theorema Egregium and geodesics — the straightest paths on a curved surface.
• Stage C — Manifolds. Curved spaces of any dimension: smooth manifolds, tangent spaces, and differential forms leading to the grand Stokes' theorem.

Stage A — Curves

1. Parametrized Curves
2. Arc Length and Reparametrization
3. Curvature and the Frenet Frame

Stage B — Surfaces

1. Parametrized Surfaces
2. The First Fundamental Form
3. The Second Fundamental Form and Curvature
4. Gaussian Curvature and Theorema Egregium
5. Geodesics

Stage C — Manifolds

1. Smooth Manifolds
2. Tangent Spaces and Vector Fields
3. Differential Forms and Stokes' Theorem

Let's get started

We begin with the simplest curved object of all — a single curve traced through space — and ask how to describe it with a moving point. Everything about bending grows from there.

Let's get started → Parametrized Curves