Math 701:

Calculus of Variations and Tensor Calculus

(Table of Contents and Its Time Line)

PART I: CALCULUS OF VARIATIONS

Lecture 1

Chapter I: Fundamental Ideas

1. Multivariable Calculus as a Prelude to Calculus of Variations
2. Some Simple Problems in Calculus of Variations
3. Methods for Solving Problems in Calculus of Variations
   1. Method of Finite Differences
      Lecture 2
   2. Method of Variations
      1. Variants and Variations
      2. Variational Derivative
      3. Euler's Differential Equation
   3. Solved Example
      Lecture 3
4. Integration of Euler's Differential Equation

Chapter II: Generalizations

1. Functional with Several Unknown Functions
   Lecture 4
2. Extremum Problems with Side Conditions
   1. Isoperimetric Problems
      1. Heuristic Solution
      2. Solution via Constraint Manifold
      Lecture 5
   2. Variational Problems with Finite Constraints
3. Variable End Point Problem
4. Extremum Problem at a Moment of Time Symmetry
   Lecture 6

   Generic Variable Endpoint Problem

   1. General Variations in the Functional
   2. Transversality Conditions
   3. Junction Conditions
   Lecture 7
   Parametrization Invariant Problem
   1. Variational Principle for a Geodesic
      Lecture 8
   2. Equation of Geodesic Motion
      1. Parametric Invariance
      2. Parametrization in Terms of Curve Length
         Lecture 9
      3. Physical Significance of the Equation for a Geodesic
      4. Equivalence Principle and ``Gravitation''=``Geometry''

Lecture 10

Chapter III: Variational Formulation of Mechanics

1. Hamilton's Principle
   Lecture 11
2. Hamilton-Jacobi Theory
   1. The Dynamical Phase
   2. Momentum and the Hamiltonian
   3. The Hamilton-Jacobi Equation
      Lecture 12
   4. Hamilton-Jacobi Description of Motion: Constructive Interference
      Lecture 13
   5. Spacetime History of a Wave Packet
   6. Hamilton's Equations of Motion
      Lecture 14
   7. Phase Space of a Dynamical System
   8. Constructive Interference Implies Extremal Paths
      Lecture 15
   9. Applications
      1. Hamilton-Jacobi Equation Relative to Curvilinear Coordinates
      2. Separation of Variables
      3. Particle Motion in a Combined Dipole-Monopole Potential
      4. Motion in the Field of Two Unequal Bodies: Prolate Spheroidal Coordinates
   Lecture 16
3. Hamilton's Principle for the Mechanics of a Continuous Medium
   1. Variational Principle
   2. Euler-Lagrange Equation
      Lecture 17
   3. Examples and Applications

Chapter IV: Direct Methods in the Calculus of Variations

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

Lecture 19

PART II: TENSOR CALCULUS

Chapter V: Tensor Algebra

1. Vectors vs. Covectors
   1. Construction of Linear Functionals: The Dual Basis
      Lecture 20

   2. The Differential of a Function
   3. Vector as a Derivation
   4. Dual Bases
   5. Invariance of Duality
   Lecture 21
2. Metric on a Vector Space
3. 1. Metric as a Map Between Vectors and Covectors
   2. Reciprocal Vector Basis
   Lecture 22
4. Tensors as Multilinear Maps
   1. Tensors: Their Components
   2. Tensors: Their Basis Representation
   3. Tensor Space
   4. How to Construct New Tensors
      1. Raising and Lowering Indeces
      2. Contraction of a Tensor

Lecture 23

Chapter VI: Geometrical Structures

1. The Commutator of Two Vector Fields
   Lecture 24
2. Parallel Transport
   1. Covariant Differential
      Lecture 25
   2. Covariant Derivative
      1. Differential of a Function
      2. Differential of a Vector
      3. Differential of a Vector Field
      4. Rules for Taking Covariant Derivatives
   3. Parallel Transport in the Euclidean Plane
   4. Covariant Derivative of a Vetor Field
   5. Covariant Derivative of a Covector Field
   6. Covariant Derivative of a Tensor Field
      Lecture 26
   7. Parallel Vector Fields
3. Torsion
   1. Closed Parallolograms
   2. Burger's Vector
   Lecture 27
4. Curvature
   1. Rotation of a Vector Transported Around a Closed Loop
   2. Riemann Curvature Tensor
   3. Equation of Geodesic Deviation
   4. Rotation = Relative Acceleration
   Lecture 28
5. Metric
   1. Metric-Induced Parallel Transport
   2. Metric Compatibility via Extremal Paths
   Lecture 29
6. Some Computational Aids
   1. Divergence of a Vector Field
   2. Gauss's Theorem

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