An Introduction To Geometrical Physics [BETTER]
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An Introduction To Geometrical Physics [BETTER]
This book enables the reader to discover elementary concepts of geometric algebra and its applications with lucid and direct explanations. Why would one want to explore geometric algebra What if there existed a universal mathematical language that allowed one: to make rotations in any dimension with simple formulas, to see spinors or the Pauli matrices and their products, to solve problems of the special theory of relativity in three-dimensional Euclidean space, to formulate quantum mechanics without the imaginary unit, to easily solve difficult problems of electromagnetism, to treat the Kepler problem with the formulas for a harmonic oscillator, to eliminate unintuitive matrices and tensors, to unite many branches of mathematical physics What if it were possible to use that same framework to generalize the complex numbers or fractals to any dimension, to play with geometry on a computer, as well as to make calculations in robotics, ray-tracing and brain science In addition, what if such a language provided a clear, geometric interpretation of mathematical objects, even for the imaginary unit in quantum mechanics Such a mathematical language exists and it is called geometric algebra. High school students have the potential to explore it, and undergraduate students can master it. The universality, the clear geometric interpretation, the power of generalizations to any dimension, the new insights into known theories, and the possibility of computer implementations make geometric algebra a thrilling field to unearth.
Some members of the geometric algebra represent geometric objects in Rn. Other members represent geometric operations on the geometric objects. Geometric algebra and its extension to geometric calculus unify, simplify, and generalize vast areas of mathematics that involve geometric ideas,including linear algebra, multivariable calculus, real analysis, complex analysis, and euclidean, noneuclidean, and projective geometry. They provide a unified mathematical language for physics (classical and quantum mechanics, electrodynamics, relativity), the geometrical aspects of computer science (e.g., graphics, robotics, computer vision), and engineering.
The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.
"My purpose here is to provide, with a minimum of mathematical machinery and in the fewest possible pages, a clear and careful explanation of the physical principles and applications of classical general relativity. The prerequisites are single variable calculus, a few basic facts about partial derivatives and line integrals, a little matrix algebra, and some basic physics. Only a bit of the algebra of tensors is used; it is developed in about a page of the text. The book is for those seeking a conceptual understanding of the theory, not computational prowess. Despite it's brevity and modest prerequisites, it is a serious introduction to the physics and mathematics of general relativity which demands careful study. The book can stand alone as an introduction to general relativity or it can be used as an adjunct to standard texts."
The previously mentioned "Geometric Algebra for Computer Science" is a good introduction that concentrates on the algebraic (not calculus related part) of GA.It does have material on GA's application to computer graphics, but the bulk of the text is just on the geometry behind GA.
The other is the use of Clifford algebras, quaternions and related ideas as a formalism for geometry and physics. This is popularized by Hestenes and is somewhat controversial, not because the math is wrong, but because it uses extra metric structure in cases where not logically required, and because of the tendency to rename known concepts and overstate the differences and advantages c