Geometric calculus - Wikipedia
In mathematics, geometric calculus extends geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to reproduce other mathematical theories including vector calculus , differential geometry , and differential forms .
Our first step in understanding geometric algebra is to construct what are called k-vectors. In this chapter we look at special kinds of k-vectors, the simple k-vectors, which have a very nice geometrical interpretation. We move in the next chapter to general k-vectors. Our presentation is rapid and superficial. Details about the construction
following fundamental concepts of Geometric Calculus, explaining their unique features and advantages: • Universal Geometric Algebra – arbitrary dimension and signature • Linear and multilinear algebra (tensors, determinants) • Vector manifolds – for representing any manifold • Directed integrals and differential forms
My purpose here is to demonstrate some of the scope and power of geometric algebra and geometric calculus. I will illustrate this for linear algebra, vector calculus, real analysis, complex analysis, and several geometries: euclidean, noneuclidean, projective, and conformal. I will also outline several applications.
All 4 of Maxwell’s equations in 1! The most compact formulation of the Maxwell equations. Unifies all four equations in one. More than some symbolic trickery. The vector derivative is an invertible operator. And extend it to arbitrary dimensions!
The purpose of these notes is to provide a limited exposure to the ideas of geometric algebra/calculus and to simultaneously use them as a vehicle to introduce the reader to some of the important concepts of multivariable analysis. (The emphasis here is on the geometric-algebraic side, and we do very little with ε−δarguments.) Our target
Geometric calculus provides an at once simpler, more general, more powerful, and easier to grasp way to break loose from R3.1Section5.4, Exact Fields, translates elementary differential forms definitions, theorems, and examples to geometric calculus. Linear algebra is the natural mathematical background for vector calculus.
My purpose is to demonstrate some of the scope and power of geometric algebra and geometric calculus. I will illustrate this for linear algebra, multivariable calculus, real analysis, complex analysis, and several geometries: euclidean, noneuclidean, projective, and conformal. I will also outline several applications.
Geometric calculus is an extension of geometric algebra to include the analytic operations of dif-ferentiation and integration. It is developed in this book as a computational language for physics. The calculus is designed for e±cient computation and representation of geometric relations.
Geometric calculus builds on the foundation of geometric algebra. It allows one to use the language of GA to do multivariable calculus in simpler ways, and often makes it possible to derive formulas without introducing an explicit coordinate system. A particularly noteworthy result in GC is the generalized