INTERDYSCYPLINARY CENTRE
FOR MATHEMATICAL AND COMPUTATIONAL MODELLING | |

**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.

**To download:** Click here
(about 275 k in .pdf format).

** 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.

- 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

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

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

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

- Flow
- Extrusion
- Lie and wedge
- The Poincare Lemma
- Poincare gauge

My address:

EdF

1 Av. Gal de Gaulle,

92141 CLAMART

FRANCE

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

- 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
- gradient
- "proxy" field Back to Contents
- normal part of a form
- Hodge operator
- grad, rot, div, and d
- grad, rot, and div on surfaces
- scalar product of forms
- functional spaces of fields and forms
- codifferential
- Laplacian
- Flow of a vector field
- Extrusion by a flow
- Lie derivative