**Chapter I: Fundamental Ideas**

- Multivariable Calculus as a Prelude to Calculus of Variations
- Some Simple Problems in Calculus of Variations
- Methods for Solving Problems in Calculus of Variations
- Method of Finite Differences
*Lecture 2* - Method of Variations
- Variants and Variations
- Variational Derivative
- Euler's Differential Equation

- Solved Example
*Lecture 3*

- Method of Finite Differences
- Integration of Euler's Differential Equation

**Chapter II: Generalizations**

- Functional with Several Unknown Functions
*Lecture 4* - Extremum Problems with Side Conditions
- Isoperimetric Problems
- Heuristic Solution
- Solution via Constraint Manifold

*Lecture 5* - Variational Problems with Finite Constraints

- Isoperimetric Problems
- Variable End Point Problem
- Extremum Problem at a Moment of Time Symmetry
*Lecture 6*Generic Variable Endpoint Problem

- General Variations in the Functional
- Transversality Conditions
- Junction Conditions

*Lecture 7*- Variational Principle for a Geodesic
*Lecture 8* - Equation of Geodesic Motion
- Parametric Invariance
- Parametrization in Terms of Curve Length
*Lecture 9* - Physical Significance of the Equation for a Geodesic
- Equivalence Principle and ``Gravitation''=``Geometry''

**Chapter III: Variational Formulation of Mechanics**

- Hamilton's Principle
*Lecture 11* - Hamilton-Jacobi Theory
- The Dynamical Phase
- Momentum and the Hamiltonian
- The Hamilton-Jacobi Equation
*Lecture 12* - Hamilton-Jacobi Description of Motion: Constructive Interference
*Lecture 13* - Spacetime History of a Wave Packet
- Hamilton's Equations of Motion
*Lecture 14* - Phase Space of a Dynamical System
- Constructive Interference Implies Extremal Paths
*Lecture 15* - Applications
- Hamilton-Jacobi Equation Relative to Curvilinear Coordinates
- Separation of Variables
- Particle Motion in a Combined Dipole-Monopole Potential
- Motion in the Field of Two Unequal Bodies: Prolate Spheroidal Coordinates

*Lecture 16* - Hamilton's Principle for the Mechanics of a Continuous Medium
- Variational Principle
- Euler-Lagrange Equation
*Lecture 17* - Examples and Applications

**Chapter IV: Direct Methods in the Calculus of Variations**

- Minimizing Sequence
- Finite-Dimensional Approximation
*Lecture 18* - Rayleigh's Variational Principle
- The Rayleigh Quotient
- The Rayleigh-Ritz Principle
- Vibrations of a Circular Membrane

**Chapter V: Tensor Algebra**

- Vectors vs. Covectors
- Construction of Linear Functionals: The Dual Basis
*Lecture 20* - The Differential of a Function
- Vector as a Derivation
- Dual Bases
- Invariance of Duality

*Lecture 21* - Construction of Linear Functionals: The Dual Basis
- Metric on a Vector Space
- Metric as a Map Between Vectors and Covectors
- Reciprocal Vector Basis

*Lecture 22*- Tensors as Multilinear Maps
- Tensors: Their Components
- Tensors: Their Basis Representation
- Tensor Space
- How to Construct New Tensors
- Raising and Lowering Indeces
- Contraction of a Tensor

**Chapter VI: Geometrical Structures**

- The Commutator of Two Vector Fields
*Lecture 24* - Parallel Transport
- Covariant Differential
*Lecture 25* - Covariant Derivative
- Differential of a Function
- Differential of a Vector
- Differential of a Vector Field
- Rules for Taking Covariant Derivatives

- Parallel Transport in the Euclidean Plane
- Covariant Derivative of a Vetor Field
- Covariant Derivative of a Covector Field
- Covariant Derivative of a Tensor Field
*Lecture 26* - Parallel Vector Fields

- Covariant Differential
- Torsion
- Closed Parallolograms
- Burger's Vector

*Lecture 27* - Curvature
- Rotation of a Vector Transported Around a Closed Loop
- Riemann Curvature Tensor
- Equation of Geodesic Deviation
- Rotation = Relative Acceleration

*Lecture 28* - Metric
- Metric-Induced Parallel Transport
- Metric Compatibility via Extremal Paths

*Lecture 29* - Some Computational Aids
- Divergence of a Vector Field
- Gauss's Theorem