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Recently, new techniques have been presented that discretize continuous elasticity variables as cochains over a primal mesh, representing the solid, and an appropriately defined dual one. Discrete strain and stress can be then thought of as a vector-valued 1-form (or vector-valued 1-cochain on the primal mesh) and covector-valued 2-form (or vector-valued 2-cochain on the dual mesh) respectively. The governing equations can be formulated by requiring energy balance and invariance under time-dependent rigid translations and rotations of the ambient space. To obtain those, we project the discrete stress into normal and tangential components and formulate the boundary value problem with a system of two matrix equations. This allow for treating both classical and coupled-stress (micro-polar) elasticity. The link between discrete strains and stresses is provided by a material discrete Hodge star operator, which we define to include geometric and physical factors, such as lengths, areas, and moduli of elasticity and rigidity. The performance of the proposed formulation is demonstrated by a simple example.

In this chapter we will introduce and discuss at length one of the ways that physicists sometimes visualize “nice” differential forms. In essence, we will be considering ways of visualizing one-forms, two-forms, and three-forms on a vector space. That is, we will find a “cartoon picture” of \(\alpha _p \in T^*_p\mathbb {R}^2\) and \(\alpha _p \in T^*_p\mathbb {R}^3\). Our picture of αp will be superimposed on the vector space \(T_p\mathbb {R}^3\). This perspective is developed extensively in Misner, Thorne, and Wheeler’s giant book Gravitation. In fact, they make some efforts to develop the four-dimensional picture (for space-time) as well, however we will primarily stick to the two and three-dimensional cases here. Section one focuses on the two-dimensional case and sections two through four focuses on the three-dimensional case. Then in section five we expand our cartoon picture to general two and three-dimensional manifolds. Again, this cartoon picture really only applies to “nice” differential forms, of the kind physicists are more likely to encounter, but it is still useful for forms and manifolds that are not overly complicated. Finally, in section six we introduce the Hodge star operator. Our visualizations of forms in three dimensions provide a nice way to visualize what the Hodge star operator does, which makes this a nice place to introduce it. Despite the power and usefulness of the way of visualizing differential forms in physics developed in this chapter, it is rarely encountered in mathematics. One of the reasons for this is that in reality it is not a completely general way of considering forms; when dealing with dimensions greater than four or with more abstract manifolds or with forms that are not “nice” in some sense it breaks down.

This note is a variation on the lecture given by the first named author at the conference celebrating the work (and sixty-fifth anniversary) of Jean-Michel Bismut. In this lecture a proof of some new cases of the Hodge conjecture for Shimura varieties uniformized by complex balls was sketched following [3]. In this note we exemplify the main ideas of the proof on real hyperbolic manifolds. The Hodge conjecture does not make sense anymore but, somewhat analogously, we prove that classes of totally geodesic submanifolds generate the cohomology groups of degree k of compact congruence n-dimensional hyperbolic manifolds “of simple type” as long as k is strictly smaller than n/3. This is a particular case of the main result of [2].

We present a mimetic spectral element method for the solution of the stationary Darcy’s problem. We show that the divergence constraint is satisfied exactly for both heterogeneous, non-isotropic, and deformed mesh problems.

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.

We describe our conjecture about the irreducible unitary representations of reductive Lie groups, in the special case of \(\mathrm{SL}(2, \mathbb{R})\).

We give a brief treatment of these topics. The crucial concept here is that of a differential k-form on a manifold. But before discussing that, we have to present a bit of exterior algebra.

This chapter mainly reviews Backlund transformations which arise in connection with elliptic–hyperbolic models. A recent application to optics is discussed in detail. Topics include a systematic method for producing Backlund transformations via the Hodge duality operator. The method is useful for the frequent cases in which the model equations can be expressed as nonlinear Hodge–Frobenius equations. A hierarchy of constraints for geometric variational problems is presented, in which the harshest constraint is that the field be conservative; the mildest weakening of that constraint is that the field be completely integrable; the weakest constraint produces maximally nonintegrable fields.