Book contents
- Frontmatter
- Contents
- Preface
- 0 The background: vector calculus
- 1 Affine spaces
- 2 Curves, functions and derivatives
- 3 Vector fields and flows
- 4 Volumes and subspaces: exterior algebra
- 5 Calculus of forms
- 6 Frobenius's theorem
- 7 Metrics on affine spaces
- 8 Isometries
- 9 Geometry of surfaces
- 10 Manifolds
- 11 Connections
- 12 Lie groups
- 13 The tangent and cotangent bundles
- 14 Fibre bundles
- 15 Connections revisited
- Bibliography
- Index
0 - The background: vector calculus
Published online by Cambridge University Press: 05 August 2012
- Frontmatter
- Contents
- Preface
- 0 The background: vector calculus
- 1 Affine spaces
- 2 Curves, functions and derivatives
- 3 Vector fields and flows
- 4 Volumes and subspaces: exterior algebra
- 5 Calculus of forms
- 6 Frobenius's theorem
- 7 Metrics on affine spaces
- 8 Isometries
- 9 Geometry of surfaces
- 10 Manifolds
- 11 Connections
- 12 Lie groups
- 13 The tangent and cotangent bundles
- 14 Fibre bundles
- 15 Connections revisited
- Bibliography
- Index
Summary
The reader of this book is assumed to have a working knowledge of vector calculus. The book is intended to explain wide generalisations of that subject. In this chapter we identify some aspects of the subject which are not always treated adequately in elementary accounts. These will be the starting points for several later developments.
Vectors
The word “vector” is used in slightly different ways in pure mathematics, on the one hand, and in applied mathematics and physics, on the other. The usefulness of the vector concept in applications is enhanced by the convention that vectors may be located at different points of space: thus a force may be represented by a vector located at its point of application. Sometimes a distinction is drawn between “free” vectors and “bound” vectors. By a free vector is meant one which may be moved about in space provided that its length and direction are kept unchanged. A bound vector is one which acts at a definite point.
In the mathematical theory of vector spaces these distinctions are unknown. In that context all vectors, insofar as they are represented by directed line segments, must originate from the same point, namely the zero vector, or origin. Only the parallelogram rule of vector addition makes sense, not the triangle rule.
Closely connected with these distinctions is a difficulty about the representation of the ordinary space of classical physics and the space-time of special relativity theory.
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- Chapter
- Information
- Applicable Differential Geometry , pp. 1 - 7Publisher: Cambridge University PressPrint publication year: 1987