Exterior Calculus: Theory and Cases

In this chapter, we introduce the main operators of Heaviside-Gibbs algebra: addition, subtraction, norm of vectors, as well as inner and cross product. From the point of view of Vector Calculus, we introduce the line and surface integrals, and the Green’s, Stokes’, and Gauss’ Theorems. The last section discusses the extension of this algebra in n-dimensional space. The examples are in plane and space.

This chapter is a review of Geometric algebra or Grassmann alge- bra on G2. This algebra is attributed to Hermann Grassmann [Die lineare Ausdehnungslehre, ein neuer Zweig der Mathematik 1842]. It has two main operators: outer product and inner product. Here, we will also study dot product, and geometric product, as well as their properties. We will start with the definition of Geometric algebra, its properties and most useful tools.With this background, we will define the differential forms in Chap. 5.

This chapter reviews and elaborates on the operators from Geometric algebra on G2 to G3. This algebra is attributed to Hermann Grassmann [Die lineare Ausdehnungslehre, ein neuer Zweig der Mathematik 1842]. It is formed by two main operators, the outer product and the inner product, it also includes the element called bivector. Here, we review their properties and their application in space.

This chapter reviews and elaborates on the operators of Geometric algebra from G3 to Gn. This algebra is attributed to Hermann Grassmann [Die lineare Ausdehnungslehre, ein neuer Zweig der Mathematik 1842]. It is formed by two main operators, the outer product and inner product. Here, a new element is introduced the multivector, we review these operators, their properties, and their application in the representation of curves, planes, and objects on space Gn.

This chapter intends to be a survey on the integration of differential forms. Here, the 0−Form, 1−Form, 2−Form, 3−Form and k−Form integrals are defined. These forms are reviewed using mapping, in particular, the cases that give rise to Simple Riemann integral, Double Riemann integral, Triple Riemann Integral, and the Line and Surface integrals. The latter two are defined in (Chap. 1), using Heaviside-Gibbs algebra.

This chapter reviews The Green’s, Stokes’, and Gauss’ Theorems as a direct result of the differentiation and integration operations set out in previous chapters. All the exercises are solved using the Grassmann algebra. The Fundamental theorem of calculus is introduced at the end of this chapter, as an extension of the theorems studied here.

This chapter gives an alternative solution to the spread of an epidemic outbreak of k dimension, using a k−Form. The k−region, derivative, and integral of this k−Form are interpreted. An extension of the k dimension is proposed using a k−Form equivalent to the electric current and the magnetic field, known as Ampere’s law. An algorithm to determine the main function of a protein is introduced using a k−Form. Finally, the k−region, derivative, and integral of this k−Form are interpreted.