Abstract
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 Poincaré 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.
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Notes
- 1.
Given an associative algebra \(\mathfrak{A}\), a bilateral ideal I is a subalgebra of \(\mathfrak{A}\) such that for any \(a,b \in \mathfrak{A}\) and for y ∈ I, ay ∈ I, yb ∈ I and ayb ∈ I. More on ideals on Chap. 3
- 2.
If we do the analogous construction of the exterior algebra using V instead of \(\boldsymbol{V } \equiv \mathbf{V}^{{\ast}}\) , then the elements of the resulting space are called multivectors.
- 3.
- 4.
- 5.
We call metric isomorphism a vector space isomorphism satisfying Eq. (2.51). The term isometry will be reserved to designate a metric isomorphism from a space onto itself.
- 6.
However, take into account that Clifford algebras \(\mathcal{C}\ell(\mathbf{V},\mathtt{\mathring{g}})\) and \(\mathcal{C}\ell(\mathbf{V},\mathtt{\mathring{g}}^{{\prime}})\) over the same vector space may be isomorphic as algebras (but not as graded algebras) even if \(\mathop{\mathtt{\mathring{g}}}\limits^{.}\) and \(\mathtt{\mathring{g}}^{{\prime}}\) do not have the same signature. The reader may verify the validity of this statement by inspecting Table 3.1 in Chap. 3 and finding examples.
- 7.
The possibility of introducing different Clifford products in the same Clifford algebra was already established by Arcuri [2]. A complete study of that issue is given in [8]. The relation between different Clifford products is an essential tool in the theory of the gravitational field as presented in [4], where this field is represented by the gauge metric extensor h (Sect. 2.8.1).
- 8.
The proofs of the propositions of this section are left to the reader.
- 9.
The signature of a metric tensor is sometimes defined as the number ( p − q).
- 10.
If the reader has any difficulty in solving that exercise he must consult, e.g., [17].
- 11.
Recall that the norm of a multiform \(X \in \bigwedge V\) is defined by \(\left \Vert X\right \Vert = \sqrt{X \cdot X}\), where the symbol ⋅ refers to the Euclidean scalar product.
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Rodrigues, W.A., Capelas de Oliveira, E. (2016). Multivector and Extensor Calculus. In: The Many Faces of Maxwell, Dirac and Einstein Equations. Lecture Notes in Physics, vol 922. Springer, Cham. https://doi.org/10.1007/978-3-319-27637-3_2
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