Vector and Geometric Calculus
Book Preface
This text, Vector and Geometric Calculus, is intended for the sophomore vector calculus course. It is a sequel to my text Linear and Geometric Algebra. That text is a prerequisite for this one. . Linear algebra and vector calculus have provided the basic vocabulary of mathematics in. dimensions greater than one for the past one hundred years. Just as geometric algebra generalizes linear algebra in powerful ways, geometric calculus generalizes vector calculus iii powerful ways.
Traditional vector calculus topics are covered here, as they , must be, since readers will encounter them in other texts and out in the world. There is a chapter on differential geometry, used today in many disciplines, including architecture, computer graphics, computer vision, econometrics, engineering, geology, image processing, and physics. ‘ Large parts of vector calculus are confined to R3 due to the extensive use of the cross product. Tensors and differential forms are two traditional formalisms used to extend to higher dimensions. Geometric calculus provides an at once simpler and more powerful way to break loose from R3.1 Appendix D provides a short comparison of differential forms and geometric calculus.
Even today it is unusual for a vector calculus text to have a linear algebra prerequisite. This has to do, I suppose, with publishers insisting that authors write to the largest possible audience. 1 use the linear algebra prerequisite to advantage. For example, I use the language of linear transformations, using matrices’only when appropriate. And since geometric algebra is at my disposal, linear transformations extend to outermorphisms. Another example: tangent spaces can be manipulated with geometric algebra operations.
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