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geometric algebra and calculus by alan bromborsky [pdf]

# An Introduction to Geometric Algebra and Calculus by Alan Bromborsky

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An Introduction to Geometric Algebra and Calculus Alan Bromborsky.
Geometric algebra is the Clifford algebra of a finite dimensional vector space over real scalars cast in a form most appropriate for physics and engineering. This was done by David Hestenes (Arizona State University) in the 1960’s. From this start he developed the geometric calculus whose fundamental theorem includes the generalized Stokes theorem, the residue theorem, and new integral theorems not realized before. Hestenes likes to say he was motivated by the fact that physicists and engineers did not know how to multiply vectors.
Researchers at Arizona State and Cambridge have applied these developments to classical mechanics,quantum mechanics, general relativity (gauge theory of gravity), projective geometry, conformal geometry, etc.

Book Detail :-
Title: An Introduction to Geometric Algebra and Calculus
Edition:
Author(s): Alan Bromborsky
Publisher:
Series:
Year: 2014
Pages: 223
Type: PDF
Language: English
ISBN:
Country:

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Book Contents :-
An Introduction to Geometric Algebra and Calculus Alan Bromborsky cover the following topics.

• Basic Geometric Algebra
Axioms of Geometric Algebra, Why Learn This Stuff?, Inner, ·, and outer, ?, product of two vectors and their basic properties, Outer, ?, product for r Vectors in terms of the geometric product, Alternate Definition of Outer, ?, product for r Vectors, Useful Relation’s, Projection Operator, Basis Blades, G (3, 0) Geometric Algebra (Euclidian Space) , G (1, 3) Geometric Algebra (Spacetime), Reflections, Rotations, Definitions, General Rotation, Euclidean Case, Minkowski Case, Expansion of geometric product and generalization of · and ?, Duality and the Pseudoscalar, Reciprocal Frames, Coordinates, Linear Transformations, Definitions, Adjoint, Inverse, Commutator Product
Examples of Geometric Algebra
Quaternions, Spinors, Geometric Algebra of the Minkowski Plane, Lorentz Transformation
Geometric Calculus - The Derivative
Definitions, Derivatives of Scalar Functions, Product Rule, Interior and Exterior Derivative, Derivative of a Multivector Function, Spherical Coordinates, Analytic Functions
Geometric Calculus - Integration
Line Integrals, Surface Integrals, Directed Integration - n-dimensional Surfaces, k-Simplex Definition, k-Chain Definition (Algebraic Topology), Simplex Notation, Fundamental Theorem of Geometric Calculus, The Fundamental Theorem At Last!, Examples of the Fundamental Theorem, Divergence and Green’s Theorems, Cauchy’s Integral Formula In Two Dimensions (Complex Plane), Green’s Functions in N-dimensional Euclidean Spaces
Geometric Calculus on Manifolds
Definition of a Vector Manifold, The Pseudoscalar of the Manifold, The Projection Operator, The Exclusion Operator, The Intrinsic Derivative, The Covariant Derivative, Coordinates and Derivatives, Riemannian Geometry, Manifold Mappings, The Fundmental Theorem of Geometric Calculus on Manifolds, Divergence Theorem on Manifolds, Stokes Theorem on Manifolds, Differential Forms in Geometric Calculus, Inner Products of Subspaces, Alternating Forms, Dual of a Vector Space, Standard Definition of a Manifold, Tangent Space, Differential Forms and the Dual Space, Connecting Differential Forms to Geometric Calculus
Multivector Calculus
New Multivector Operations, Derivatives With Respect to Multivectors, Calculus for Linear Functions
Multilinear Functions (Tensors)
Algebraic Operations, Covariant, Contravariant, and Mixed Representations, Contraction, Differentiation, From Vector/Multivector to Tensor, Parallel Transport Definition and Example, Covariant Derivative of Tensors, Coefficient Transformation Under Change of Variable
Lagrangian and Hamiltonian Methods
Lagrangian Theory for Discrete Systems, The Euler-Lagrange Equations, Symmetries and Conservation Laws, Examples of Lagrangian Symmetries, Lagrangian Theory for Continuous Systems, The Euler Lagrange Equations, Symmetries and Conservation Laws, Space-Time Transformations and their Conjugate Tensors, Case 1 - The Electromagnetic Field, Case 2 - The Dirac Field, Case 3 - The Coupled Electromagnetic and Dirac Fields
Lie Groups as Spin Groups
Introduction, Simple Examples, SO (2) - Special Orthogonal Group of Order 2, GL (2, <) - General Real Linear Group of Order 2, Properties of the Spin Group, Every Rotor is the Exponential of a Bivector, Every Exponential of a Bivector is a Rotor, The Grassmann Algebra, The Dual Space to Vn, The Mother Algebra, The General Linear Group as a Spin Group, Endomorphisms of < n
Classical Electromagnetic Theory
Maxwell Equations, Relativity and Particles, Lorentz Force Law, Relativistic Field Transformations, The Vector Potential, Radiation from a Charged Particle
Further Properties of the Geometric Product, BAC-CAB Formulas, Reduction Rules for Scalar Projections of Multivectors, Curvilinear Coordinates via Matrices, Practical Geometric Calculus on Manifolds, Direct Sum of Vector Spaces, sympy/galgebra evaluation of GL (n, <) structure constants, Blade Orientation Theorem, Case Study of a Manifold for a Model Universe, The Edge of Known Space

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