Introduction

Why Functional Differential Geometry?

Differential geometry is the mathematical language of general relativity, gauge theory, and modern theoretical physics. But it is notorious for its dense notation — indices, summation conventions, and abstract tensor calculus that obscure the underlying ideas.

Gerald Jay Sussman and Jack Wisdom (authors of Structure and Interpretation of Classical Mechanics and creators of Scheme) wrote Functional Differential Geometry to fix this. Their insight: by programming the mathematics, you are forced to be precise. A computer will not tolerate vague or incomplete constructions. Every ambiguity in the notation must be resolved before you can write working code.

This course follows their approach, translated into Python. Every concept — derivative, tangent vector, metric, curvature — is represented as a Python function or higher-order function. You will implement differential geometry from scratch, building up from function composition to geodesics on the sphere.

What You Will Learn

This course contains 15 lessons organized into 4 chapters:

  1. Functional Foundations — Function composition, the derivative operator D as a higher-order function, and partial derivatives as operators.
  2. Manifolds — Coordinate systems as pairs of inverse functions (chart/point), the Jacobian of coordinate transformations, and tangent vectors as derivations.
  3. Vector & Form Fields — Vector fields as differential operators, one-form fields as duals of vector fields, the exterior derivative (curl), and the Lie bracket.
  4. Connections & Curvature — The metric tensor, Christoffel symbols from the metric, parallel transport along curves, the Riemann curvature tensor, and geodesic equations.

By the end, you will have implemented the mathematical machinery needed to describe curved spacetime — the foundation of general relativity.

Prerequisites

  • Basic Python (functions, lists, loops)
  • Calculus (derivatives, partial derivatives)
  • Linear algebra (vectors, matrices) is helpful

The Book

This course is based on Functional Differential Geometry by Gerald Jay Sussman and Jack Wisdom (MIT Press, 2013). The original uses Scheme; we use Python. The ideas are identical.

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