 INTERDYSCYPLINARY CENTRE FOR MATHEMATICAL AND COMPUTATIONAL MODELLING Applied differential geometry

A compendium

by Alain Bossavit

Readership: Applied mathematicians, physicists, engineers concerned with partial differential equations of classical physics.

Requisites: Linear algebra, calculus, especially vector analysis, some familiarity with analytical geometry. The text is not tailored as a first course, and some previous exposition to tensor calculus is assumed.

Motivation: Shortly speaking, differential geometry offers a valuable alternative to vector analysis. Some will tell you that everything one can do with differential forms can be done with vector fields, and this is true -- in spatial dimension 3, that is. But the converse also is true, and much more fun. As a bonus, using forms sheds light on otherwise puzzling analogies or coincidences: Why and how, for instance, - div o grad and rot o rot can behave in similar ways.

What's in there:

This document is what "compendium" suggests: Not a tutorial, but a list, in logical order, of basic concepts of differential geometry, each with a condensed description. (Click here to view the list of defined terms.) Emphasis is put on concepts that have relevance to boundary-value problems of classical physics.

1. Differentiable manifolds and their denizens

Manifolds, charts, atlases
Manifolds with boundary
Tangent vectors
Tangent space at a point; at a boundary point; transverse vectors
Tangent map
Immersions, embeddings
Diffeomorphisms
Basis vectors at a point, coordinate lines
Tangent bundle, fibers, fibered spaces
Vector fields, as derivations
Frames, local and global
Holonomy
Covectors, cotangent bundle, p-covectors
Exterior product (wedge product)
Inner product
Differential forms
Exterior product and inner product of forms
Trace of a form
Tensors

2. Orientation and twisted forms

Volume
Orientable manifold
Local volumes, direct frames
Inner orientation
"Ambient" space, outer orientation
Twisted covectors
Twisted differential forms, or "densities"

3. Integration and the Stokes theorem

Reference simplex, simplicial cells
Triangulation, simplicial mesh
Mesh refinement, meaning of "m --> 0"
Integral of a density
Exterior differential (operator d)
Stokes
Lie derivative

4. Metric structures on manifolds

Metric, Riemannian manifolds
Hodge operator
Scalar products between p-forms
Codifferential and Laplace operator
Integration by parts

5. The Lie derivative

Flow
Extrusion
Lie and wedge
The Poincare Lemma
Poincare gauge

Reference list

EdF
1 Av. Gal de Gaulle,
92141 CLAMART
FRANCE

My email address: alain.bossavit@der.edfgdf.fr

Page created December 6, 2001. Last modified September 13, 2002 © A. Bossavit

List of defined terms and concepts

affine space
dimension
barycenter
affine basis
affine subspace
barycentric coordinates
domain
codomain
manifold
dimension
compatibility of charts
atlas
complete atlas
manifold with boundary
tangent vectors
tangent space
transverse vector (outward, inward)
tangent map
immersion
diffeomorphism
embedding
basis vectors
coordinate lines          Back to Contents
tangent bundle
bundle, projection, fibers, fibered space
vector fields
components
frame, local and global
holonomy
covector
cotangent bundle
alternating map
exterior product, wedge product
inner product
multivector
differential form
scalar fields
tensors
Einstein convention
exterior product
inner product
trace
orientation
volumes, local volumes
direct frames, skew frames
orientable manifold, oriented manifold
inner orientation
induced orientation
ambient manifold
outer orientation
hypersurface
crossing direction          Back to Contents
twisted covectors
twisted vectors
axial vectors
differential forms
densities
trace of a form
reference simplex
faces
simplicial map
simplicial cell
triangulation
simplicial tiling
simplicial mesh
refinement, uniform refinement
integral
exterior differential
Stokes's theorem.
integration by parts
Lie derivative, convective derivative
divergence of a vector field
metric, distance
Riemannian manifold
Euclidean space
connexion
sharp and flat operators
covariant components
norm of a covector
orthonormal basis