Differential Geometry of Manifolds

Stephen Lovett is an associate professor of mathematics at Wheaton College in Illinois. Lovett has also taught at Eastern Nazarene College and has taught introductory courses on differential geometry for many years. Lovett has traveled extensively and has given many talks over the past several years on differential and algebraic geometry, as well as cryptography.

Analysis of Multivariable Functions
Functions from Rn to Rm
Continuity, Limits, and Differentiability
Differentiation Rules: Functions of Class Cr
Inverse and Implicit Function Theorems
Coordinates, Frames, and Tensor Notation
Curvilinear Coordinates
Moving Frames in Physics
Moving Frames and Matrix Functions
Tensor Notation
Differentiable Manifolds
Definitions and Examples
Differentiable Maps between Manifolds
Tangent Spaces and Differentials
Immersions, Submersions, and Submanifolds
Chapter Summary
Analysis on Manifolds
Vector Bundles on Manifolds
Vector Fields on Manifolds
Differential Forms
Integration on Manifolds
Stokes’ Theorem
Introduction to Riemannian Geometry
Riemannian Metrics
Connections and Covariant Differentiation
Vector Fields Along Curves: Geodesics
The Curvature Tensor
Applications of Manifolds to Physics
Hamiltonian Mechanics
Electromagnetism
Geometric Concepts in String Theory
A Brief Introduction to General Relativity
Point Set Topology
Introduction
Metric Spaces
Topological Spaces
Proof of the Regular Jordan Curve Theorem
Simplicial Complexes and Triangulations
Euler Characteristic
Calculus of Variations
Formulation of Several Problems
The Euler-Lagrange Equation
Several Dependent Variables
Isoperimetric Problems and Lagrange Multipliers
Multilinear Algebra
Direct Sums
Bilinear and Quadratic Forms
The Hom Space and the Dual Space
The Tensor Product
Symmetric Product and Alternating Product
The Wedge Product and Analytic Geometry