Applied Differential Geometry by Burke

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• Author: William Burke
• Title: Applied Differential Geometry
• Amazon Link: https://www.amazon.com/dp/0521269296/?tag=pfamazon01-20
• Prerequisities:
• Level: Undergrad

Table of Contents:

[LIST]
[*] Preface
[*] Glossary of notation
[*] Introduction
[*] Tensor in linear spaces
[LIST]
[*] Linear and affine spaces
[*] Differential calculus
[*] Tensor algebra
[*] Alternating products
[*] Special relativity
[*] The uses of covariance
[/LIST]
[*] Manifolds
[LIST]
[*] Manifolds
[*] Tangent vectors and 1-forms
[*] Lie bracket
[*] Tensors on manifolds
[*] Mappings
[*] Cotangent bundle
[*] Tangent bundle
[*] Vector fields and dynamical systems
[*] Contact bundles
[*] The geometry of thermodynamics
[/LIST]
[*] Transformations
[LIST]
[*] Lie groups
[*] Lie derivative
[*] Holonomy
[*] Contact transformations
[*] Symmetries
[/LIST]
[*] The calculus of differential forms
[LIST]
[*] Differential forms
[*] Exterior calculus
[*] The * Operator
[*] Metric Symmetries
[*] Normal forms
[*] Index notation
[*] Twisted differential forms
[*] Integration
[*] Cohomology
[/LIST]
[*] Applications of the exterior calculus
[LIST]
[*] Diffusion equations
[*] First-order partial differential equations
[*] Conservation laws
[*] Calculus of variations
[*] Constrained variations
[*] Variations of multiple integrals
[*] Holonomy and thermodynamics
[*] Exterior differential systems
[*] Symmetries and similarity solutions
[*] Variational principles and conservation laws
[*] When not to use forms
[/LIST]
[*] Classical electrodynamics
[LIST]
[*] Electrodynamics and differential forms
[*] Electrodynamics in spacetime
[*] Laws of conservation and balance
[*] Maccroscopic electrodynamics
[*] Electrodynamics of moving bodies
[/LIST]
[*] Dynamics of particles and fields
[LIST]
[*] Lagrangian mechanics of conservative systems
[*] Lagrange's equations for general systems
[*] Lagrangian field theory
[*] Hamiltonian systems
[*] Symplectic geometry
[*] Hamiltonian optics
[*] Dynamics of wave packets
[/LIST]
[*] Calculus on fiber bundles
[LIST]
[*] Connections
[*] Parallel transport
[*] Curvature and torsion
[*] Covariant differentiation
[*] Metric connections
[/LIST]
[*] Gravitations
[LIST]
[*] General relativity
[*] Geodesics
[*] Geodesic deviation
[*] Symmetries and conserved quantities
[*] Schwarzschild orbit problem
[*] Light deflection
[*] Gravitational lenses
[*] Moving frames
[/LIST]
[*] Bibliography
[*] Index
[/LIST]

 

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One of the few differential geometry books that emphasize the importance of twisted tensors.