Exterior calculus

The exterior calculus generalizes vector calculus on Euclidean spaces to smooth manifolds. Its basic concepts include differential forms, the exterior derivative, and Hodge duality.

Exterior calculus is a standard topic in differential geometry. It is covered in most textbooks on the subject and also in books with names like “mathematics for physcists.” Here are a few that I like:

Baez & Muniain, 1994: Gauge fields, knots, and gravity (doi, errata , solutions )
- Part I is a crash course in differential geometry and exterior calculus in a modern, coordinate-free style, culminating in a reformulation of Maxwell’s equations using the exterior derivative and codifferential
- Notes to chapters give references for further reading, in particular: Warner, 1983: Foundations of differentiable manifolds and Lie groups
Lee, 2009: Manifolds and differential geometry (doi)
- Theorem 2.72 gives the equivalence between the algebraic definition of vector fields as global derivations, used by Baez & Muniain, and the usual definition of vector fields as smooth sections of the tangent bundle

Twisted differential forms

Differential forms can be integrated only over oriented submanifolds. Another kind of differential form, called a pseudoform or twisted form , can be integrated over submanifolds that are pseudo-oriented (aka, outer oriented or transversely oriented). For emphasis, ordinary differential forms may then be called straight or untwisted forms. de Rham calls straight and twisted forms even and odd forms, respectively. In the discrete exterior calculus, straight and twisted smooth forms correspond to primal and dual discrete forms.

Despite being invented by a mathematician—de Rham himself—twisted differential forms have mostly been ignored in pure mathematics. Some exceptions in the literature are:

Bott & Tu, 1982: Differential forms in algebraic topology, Section I.7: The nonorientable case
de Rham, 1984: Differential manifolds: Forms, currents, harmonic forms (doi), Section 5: Differential forms of odd type
Ramanan, 2005: Global calculus (doi), Section 3.3.2: Twisted forms

Twisted forms receive more attention from applied mathematicians and physicists because straight and twisted forms model different kinds of physical quantities.

Burke, 1985: Applied differential geometry, esp. Section 28: Twisted differential forms
- Unclear definitions and little rigor, but useful for geometric intuition and physical applications
- Burke, 1995, draft: Div, grad, curl are dead (pdf)
- Burke, 1995, draft: Twisted forms as they should be (ps )
Bossavit, 2005: Discretization of electromagnetic problems: The “generalized finite differences” approach (doi)
- Bossavit, 1998: On the geometry of electromagnetism (1 ,2 ,3 ,4 )
Frankel, 2012: The geometry of physics, 3rd ed., Section 2.8: Orientation and pseudoforms, Section 3.4: Integration of pseudoforms, and Section 3.5d: Forms and pseudoforms
- In the context of Maxwell’s equations, suggests that “there is a general rule of thumb concerning forms versus pseudoforms; a form measures an intensity whereas a pseudoform measures a quantity”

Densities

A top-dimensional twisted form, i.e., a twisted \(n\)-form on an \(n\)-manifold, is also called a density (cf. Wikipedia ). The nLab describes a generalization to absolute differential forms . Densities and absolute forms can be integrated over any manifolds, oriented or not.

Mathematical literature on densities:

Lee, 2012: Introduction to smooth manifolds, 2nd ed. (doi), Chapter 16, section on densities
Nicolaescu, 2021: Lectures on the geometry of manifolds, 3rd ed. (pdf)

Generalizations of exterior calculus

Aspects of exterior calculus can be extended from manifolds to generalized “smooth spaces” such as diffeological spaces.

Christensen & Wu, 2020: Exterior bundles in diffeology (arxiv)