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Math 5451:
Calculus of Variations and Tensor Calculus
(Table of Contents and Its Time Line)
PART I: CALCULUS OF VARIATIONS
Lecture 1
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
 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
 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
Parametrization Invariant Problem
 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''
Lecture 10
Chapter III: Variational Formulation of Mechanics
 Hamilton's Principle
Lecture 11
 HamiltonJacobi Theory
 The Dynamical Phase
 Momentum and the Hamiltonian
 The HamiltonJacobi Equation
Lecture 12
 HamiltonJacobi 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
 HamiltonJacobi Equation Relative to Curvilinear
Coordinates
 Separation of Variables
 Particle Motion in a Combined DipoleMonopole
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
 EulerLagrange Equation
Lecture 17
 Examples and Applications
Chapter IV: Direct Methods in the Calculus of Variations
 Minimizing Sequence
 FiniteDimensional Approximation
Lecture 18
 Rayleigh's Variational Principle
 The Rayleigh Quotient
 The RayleighRitz Principle
 Vibrations of a Circular Membrane
Lecture 19
PART II: TENSOR CALCULUS
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
 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
Lecture 23
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
 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
 MetricInduced Parallel Transport
 Metric Compatibility via Extremal Paths
Lecture 29
 Some Computational Aids
 Divergence of a Vector Field
 Gauss's Theorem
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gerlach@math.ohiostate.edu
20030711