STARLIGHT
THEORY

About frogs and hyperstates

A collection of poems and essays about frogs and hyperstates. The book is a collection of poems and essays about frogs and hyperstates. It is a collection of poems and essays about frogs and hyperstates.

Smooth Manifolds #1 medium

Quick introduction to manifolds and smooth structures above them, following the Introduction to Smooth Manifolds book by John M. Lee.

Smooth Manifolds #10 easy

Explore the fascinating world of de Rham cohomology, a powerful tool that connects differential forms to the topology of manifolds. Learn how this theory unifies various theorems in vector calculus and reveals deep geometric insights.

Smooth Manifolds #11 hard

Discover the elegant world of symplectic geometry, the mathematical framework behind classical mechanics. Learn how this geometric perspective provides deep insights into the nature of physical systems and their evolution.

Smooth Manifolds #12 hard

Enter the beautiful world of complex manifolds, where complex analysis meets differential geometry. Discover how these structures naturally arise in mathematics and physics, and learn about their special properties that make them so important.

Smooth Manifolds #13 easy

Explore one of the most profound results in mathematics: the Atiyah-Singer index theorem. Learn how this remarkable theorem connects analysis, topology, and geometry, leading to deep insights about the structure of manifolds.

Smooth Manifolds #2 medium

Quick introduction to manifolds and smooth structures above them, following the Introduction to Smooth Manifolds book by John M. Lee. Quick introduction to manifolds and smooth structures above them, following the Introduction to Smooth Manifolds book by John M. Lee.

Smooth Manifolds #3 easy

Quick introduction to manifolds and smooth structures above them, fo

Smooth Manifolds #4 hard

Quick introduction to manifolds and smooth structures above them, following the Introduction to Smooth Manifolds book by John M. Lee.

Smooth Manifolds #5 medium

Deep dive into the concept of tangent spaces on smooth manifolds, exploring how tangent vectors arise naturally as derivations on smooth functions. We'll see how these concepts generalize our intuition from calculus in ℝⁿ.

Smooth Manifolds #6 hard

Discover the elegant world of differential forms, the fundamental objects that allow us to integrate over manifolds. Learn how these mathematical objects generalize the notion of integration from calculus and provide a powerful framework for understanding geometry.

Smooth Manifolds #7 easy

Explore vector fields on smooth manifolds and their integral curves. Learn how vector fields generate flows, and discover the deep connection between vector fields and ordinary differential equations on manifolds.

Smooth Manifolds #9 hard

Learn how to measure distances and angles on manifolds using Riemannian metrics. Discover how these mathematical structures allow us to do geometry on curved spaces, generalizing our intuition from Euclidean geometry.