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Geometric Algebra for Electrical Engineers
This book introduces the fundamentals of geometric algebra and calculus, and applies those tools to the study of electromagnetism. Geometric algebra provides a structure that can represent oriented point, line, plane, and volume segments. Vectors, which can be thought of as a representation of oriented line segments, are generalized to multivectors. A full fledged, but non-commutative (i.e. order matters) mul- tiplication operation will be defined for products of vectors. Namely, the square of a vector is the square of its length. This simple rule, along with a requirement that we can sum vectors and their products, essentially defines geometric algebra. Such sums of scalars, vectors and vec...
Exploring physics with Geometric Algebra
This is an exploratory collection of notes containing worked examples of a number of applications of Geometric Algebra (GA), also known as Clifford Algebra. This writing is focused on undergraduate level physics concepts, with a target audience of somebody with an undergraduate engineering background (i.e. me at the time of writing.) These notes are more journal than book. You'll find lots of duplication, since I reworked some topics from scratch a number of times. In many places I was attempting to learn both the basic physics concepts as well as playing with how to express many of those concepts using GA formalisms. The page count proves that I did a very poor job of weeding out all the du...
Relativistic Electrodynamics
These are my personal lecture notes for the Spring 2011, University of Toronto, Relativistic Electrodynamics course (PHY450H1S). This class was taught by Prof. Erich Poppitz, with Simon Freedman handling tutorials (which were excellent lecture style lessons). Official course description: Special Relativity, four-vector calculus and relativistic notation, the relativistic Maxwell's Equations, electromagnetic waves in vacuum and conducting and non-conducting materials, electromagnetic radiation from point charges and systems of charges. This document contains a few things * My lecture notes. Typos and errors are probably mine (Peeter), and no claim nor attempt of spelling or grammar correctnes...
Condensed Matter Physics
This document is based on my lecture notes for the Winter 2013, University of Toronto Condensed Matter Physics course (PHY487H1F), taught by Prof. Stephen Julian. Official course description: “Introduction to the concepts used in the modern treatment of solids. The student is assumed to be familiar with elementary quantum mechanics. Topics include: bonding in solids, crystal structures, lattice vibrations, free electron model of metals, band structure, thermal properties, magnetism and superconductivity (time permitting)” This document contains: • Plain old lecture notes. These mirror what was covered in class, possibly augmented with additional details. • Personal notes exploring details that were not clear to me from the lectures, or from the texts associated with the lecture material. • Some worked problems attempted as course prep, for fun, or for test preparation, or post test reflection. • Links to Mathematica workbooks associated with this course.
Geometric Algebra for Computer Science
Until recently, almost all of the interactions between objects in virtual 3D worlds have been based on calculations performed using linear algebra. Linear algebra relies heavily on coordinates, however, which can make many geometric programming tasks very specific and complex-often a lot of effort is required to bring about even modest performance enhancements. Although linear algebra is an efficient way to specify low-level computations, it is not a suitable high-level language for geometric programming. Geometric Algebra for Computer Science presents a compelling alternative to the limitations of linear algebra. Geometric algebra, or GA, is a compact, time-effective, and performance-enhanc...
Basic Statistical Mechanics
This document is based on my lecture notes for the Winter 2013, University of Toronto Basic Statistical Mechanics course (PHY452H1S), taught by Prof. Arun Paramekanti. Official course description: “Classical and quantum statistical mechanics of noninteracting systems; the statistical basis of thermodynamics; ensembles, partition function; thermodynamic equilibrium; stability and fluctuations; formulation of quantum statistics; theory of simple gases; ideal Bose and Fermi systems.” This document contains: • Plain old lecture notes. These mirror what was covered in class, possibly augmented with additional details. • Personal notes exploring details that were not clear to me from the lectures, or from the texts associated with the lecture material. • Assigned problems. two problem sets. • Some worked problems attempted as course prep, for fun, or for test preparation, or post test reflection. • Links to Mathematica workbooks associated with these notes.
Clifford Algebra to Geometric Calculus
Matrix algebra has been called "the arithmetic of higher mathematics" [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebra' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quaternions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.
A New Approach to Differential Geometry using Clifford's Geometric Algebra
Differential geometry is the study of the curvature and calculus of curves and surfaces. A New Approach to Differential Geometry using Clifford's Geometric Algebra simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space. Complete with chapter-by-chapter exercises, an overview of general relativity, and brief biographies of historical figures, this comprehensive textbook presents a valuable introduction to differential geometry. It will serve as a useful resource for upper-level undergraduates, beginning-level graduate students, and researchers in the algebra and physics communities.
Understanding Geometric Algebra for Electromagnetic Theory
This book aims to disseminate geometric algebra as a straightforward mathematical tool set for working with and understanding classical electromagnetic theory. It's target readership is anyone who has some knowledge of electromagnetic theory, predominantly ordinary scientists and engineers who use it in the course of their work, or postgraduate students and senior undergraduates who are seeking to broaden their knowledge and increase their understanding of the subject. It is assumed that the reader is not a mathematical specialist and is neither familiar with geometric algebra or its application to electromagnetic theory. The modern approach, geometric algebra, is the mathematical tool set we should all have started out with and once the reader has a grasp of the subject, he or she cannot fail to realize that traditional vector analysis is really awkward and even misleading by comparison. Professors can request a solutions manual by email: pressbooks@ieee.org
Geometrical Vectors
Every advanced undergraduate and graduate student of physics must master the concepts of vectors and vector analysis. Yet most books cover this topic by merely repeating the introductory-level treatment based on a limited algebraic or analytic view of the subject. Geometrical Vectors introduces a more sophisticated approach, which not only brings together many loose ends of the traditional treatment, but also leads directly into the practical use of vectors in general curvilinear coordinates by carefully separating those relationships which are topologically invariant from those which are not. Based on the essentially geometric nature of the subject, this approach builds consistently on students' prior knowledge and geometrical intuition. Written in an informal and personal style, Geometrical Vectors provides a handy guide for any student of vector analysis. Clear, carefully constructed line drawings illustrate key points in the text, and problem sets as well as physical examples are provided.
