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Differential Geometry

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Differential Geometry

Curves, surfaces, and the geometry of manifolds

145Modules

8Units

Being rebuilt

This course is still in our original format. All the content is here and free to use, but it has not yet been through the rebuild that gives our newer courses their interactive animations and cleaner visuals — so you may hit some rough edges. Spot an error? Send us a note and we will fix it.

Recommended preparation

Multivariable calculus (Calculus 3) and linear algebra are the key tools you will draw on.

Calculus 3 Linear Algebra (Matrices)

What is Differential Geometry?INTRO

Geometry Meets CalculusINTRO

Curves and MotionINTRO

Parametric CurvesLESSON

Tangent VectorsLESSON

Regular CurvesLESSON

Arc Length ParametrizationLESSON

Curves PracticePRACTICE

Curvature of CurvesLESSON

Curvature CalculationsEXAMPLE

Curve VisualizerACTIVITY

Curvature and Torsion PracticePRACTICE

The Frenet FrameLESSON

Frenet Frame ExamplesEXAMPLE

Frenet Frame ExplorerACTIVITY

Frenet-Serret FormulasLESSON

Fundamental Theorem for CurvesLESSON

Roller Coaster DesignREAL WORLD

Curves SummarySUMMARY

Curves ReferenceREFERENCE

Curves QuizQUIZ

What are Surfaces?INTRO

Surfaces Around UsINTRO

Parametric SurfacesLESSON

Surface Parametrization ExamplesEXAMPLE

Surfaces PracticePRACTICE

Regular SurfacesLESSON

Tangent PlanesLESSON

Tangent Plane VisualizerACTIVITY

Normal VectorsLESSON

First Fundamental FormLESSON

First Fundamental Form ExamplesEXAMPLE

First Fundamental Form PracticePRACTICE

Surface AreaLESSON

Surface PlotterACTIVITY

Orientation of SurfacesLESSON

Curves on SurfacesLESSON

Surfaces SummarySUMMARY

Surfaces ReferenceREFERENCE

Surfaces QuizQUIZ

How Surfaces CurveINTRO

Intrinsic vs ExtrinsicINTRO

The Shape OperatorLESSON

Second Fundamental FormLESSON

Principal CurvaturesLESSON

Shape Operator PracticePRACTICE

Principal DirectionsLESSON

Gaussian CurvatureLESSON

Gaussian Curvature CalculationsEXAMPLE

Mean CurvatureLESSON

Surface Curvature VisualizerACTIVITY

Soap Films and Mean CurvatureREAL WORLD

Gaussian and Mean Curvature PracticePRACTICE

Umbilical PointsLESSON

Euler's Formula for CurvatureLESSON

Classification by CurvatureLESSON

Surface Curvature ExamplesEXAMPLE

Surface Curvature SummarySUMMARY

Curvature ReferenceREFERENCE

Surface Curvature QuizQUIZ

What is Intrinsic Geometry?INTRO

The Flatland PerspectiveINTRO

Christoffel SymbolsLESSON

Covariant DerivativeLESSON

Parallel TransportLESSON

Parallel Transport ExamplesEXAMPLE

Parallel Transport VisualizerACTIVITY

GeodesicsLESSON

Geodesics PracticePRACTICE

Geodesic EquationsLESSON

Geodesic CalculationsEXAMPLE

Geodesic TracerACTIVITY

The Exponential MapLESSON

Navigation on EarthREAL WORLD

IsometriesLESSON

Local IsometriesLESSON

Theorema EgregiumLESSON

Intrinsic Geometry PracticePRACTICE

Intrinsic Geometry SummarySUMMARY

Intrinsic Geometry ReferenceREFERENCE

Intrinsic Geometry QuizQUIZ

A Remarkable TheoremINTRO

Geodesic CurvatureLESSON

Geodesic Curvature PracticePRACTICE

Geodesic TrianglesLESSON

Angle ExcessLESSON

Local Gauss-BonnetLESSON

Local to GlobalINTRO

Euler Characteristic of SurfacesLESSON

Topological InvariantsLESSON

Global Gauss-Bonnet TheoremLESSON

Gauss-Bonnet for SpheresEXAMPLE

Gauss-Bonnet for the TorusEXAMPLE

Gauss-Bonnet CalculatorACTIVITY

Gauss-Bonnet PracticePRACTICE

Applications of Gauss-BonnetLESSON

Topology and CurvatureREAL WORLD

Gauss-Bonnet SummarySUMMARY

Gauss-Bonnet ReferenceREFERENCE

Gauss-Bonnet QuizQUIZ

What are Manifolds?INTRO

Beyond SurfacesINTRO

Topological ManifoldsLESSON

Charts and AtlasesLESSON

Manifold Atlas BuilderACTIVITY

Smooth StructureLESSON

Manifold Construction ExamplesEXAMPLE

Manifolds PracticePRACTICE

Smooth Maps Between ManifoldsLESSON

Tangent Spaces on ManifoldsLESSON

Tangent Space ExamplesEXAMPLE

Tangent Spaces PracticePRACTICE

The Differential of a MapLESSON

The Tangent BundleLESSON

Vector FieldsLESSON

SubmanifoldsLESSON

Manifolds SummarySUMMARY

Manifolds ReferenceREFERENCE

Manifolds QuizQUIZ

What is Riemannian Geometry?INTRO

Measuring on ManifoldsINTRO

Riemannian MetricsLESSON

The Metric TensorLESSON

Riemannian Metric ExamplesEXAMPLE

Metric ExplorerACTIVITY

Riemannian Metrics PracticePRACTICE

Length and AnglesLESSON

Levi-Civita ConnectionLESSON

Riemann Curvature TensorLESSON

Curvature Tensor CalculationsEXAMPLE

Ricci Tensor and Scalar CurvatureLESSON

Sectional CurvatureLESSON

Curvature Tensors PracticePRACTICE

Geodesics on Riemannian ManifoldsLESSON

Geodesic CompletenessLESSON

General Relativity PreviewREAL WORLD

Riemannian Geometry SummarySUMMARY

Riemannian Geometry ReferenceREFERENCE

Riemannian Geometry QuizQUIZ

Problem Solving GuideSUMMARY

Differential Geometry Master ReferenceREFERENCE

Complete Formula SheetFORMULA SHEET

Notation GuideREFERENCE

Comprehensive Differential Geometry TestQUIZ