Multivector and Extensor Calculus

This chapter is dedicated to a thoughtful exposition of the multiform and extensor calculus. Starting from the tensor algebra of a real n-dimensional vector space \(\boldsymbol{V }\) we construct the exterior algebra \(\bigwedge \boldsymbol{V }\) of \(\boldsymbol{V }\). Equipping \(\boldsymbol{V }\) with a metric tensor \(\mathring{g}\) we introduce the Grassmann algebra and next the Clifford algebra \(\mathcal{C}\ell(\boldsymbol{V },\mathit{\mathring{g}})\) associated to the pair \((\boldsymbol{V },\mathit{\mathring{g}})\). The concept of Hodge dual of elements of \(\bigwedge \boldsymbol{V }\) (called nonhomogeneous multiforms) and of \(\mathcal{C}\ell(\boldsymbol{V },\mathit{\mathring{g}})\) (also called nonhomogeneous multiforms or Clifford numbers) is introduced, and the scalar product and operations of left and right contractions in these structures are defined. Several important formulas and “tricks of the trade” are presented. Next we introduce the concept of extensors which are multilinear maps from p subspaces of \(\bigwedge \boldsymbol{V }\) to q subspaces of \(\bigwedge \boldsymbol{V }\) and study their properties. Equipped with such concept we study some properties of symmetric automorphisms and the orthogonal Clifford algebras introducing the gauge metric extensor (an essential ingredient for theories presented in other chapters). Also, we define the concepts of strain, shear and dilation associated with endomorphisms. A preliminary exposition of the Minkoswski vector space is given and the Lorentz and Poincare groups are introduced. In the remaining of the chapter we give an original presentation of the theory of multiform functions of multiform variables. For these objects we define the concepts of limit, continuity and differentiability. We study in details the concept of directional derivatives of multiform functions and solve several nontrivial exercises to clarify how to work with these notions, which in particular are crucial for the formulation of Chap. 8 which deals with a Clifford algebra Lagrangian formalism of field theory in Minkowski spacetime.