Book contents
- Frontmatter
- Contents
- Preface
- Glossary of notation
- Introduction
- I Tensors in linear spaces
- II Manifolds
- III Transformations
- IV The calculus of differential forms
- V Applications of the exterior calculus
- VI Classical electrodynamics
- VII Dynamics of particles and fields
- VIII Calculus on fiber bundles
- IX Gravitation
- Bibliography
- Index
IV - The calculus of differential forms
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Glossary of notation
- Introduction
- I Tensors in linear spaces
- II Manifolds
- III Transformations
- IV The calculus of differential forms
- V Applications of the exterior calculus
- VI Classical electrodynamics
- VII Dynamics of particles and fields
- VIII Calculus on fiber bundles
- IX Gravitation
- Bibliography
- Index
Summary
This chapter develops a set of tools for manipulating differential forms. This calculus of differential forms is the promised generalization of ordinary vector calculus. It carries to manifolds such basic notions as gradient, curl, and integral.
In this chapter, I will develop these tools. In the next chapter, I will present some simple applications of them. The true value of these tools will be seen in Chapter VI, where they are applied to electrodynamics, and in the following chapters on mechanics and gravitation. Some specialized tools only make sense when seen in the context of a specific problem. These have been deferred. Thus the concept of an exterior system waits until we have some partial differential equations to play with. These tools of the calculus of differential forms are so efficient that we now go to some lengths to cast problems into a form amenable to these methods.
Differential forms
The calculus of differential forms, often called exterior calculus, is an efficient system for manipulating functions, vectors, and differential forms, both ordinary and twisted. It is natural and very efficient for calculations, especially ones that do not involve a metric or a covariant derivative. In a later section we will bring Riemannian geometry into the exterior calculus by using the method of moving frames. The exterior calculus is particularly efficient for computations in which tensors are represented explicitly rather than as indexed arrays.
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- Chapter
- Information
- Applied Differential Geometry , pp. 147 - 206Publisher: Cambridge University PressPrint publication year: 1985