By G. N. Watson

ISBN-10: 1603861041

ISBN-13: 9781603861045

An Unabridged, Digitally Enlarged Printing, to incorporate: The Tabulation Of Bessel capabilities - Bibliography - Index Of Symbols - record Of Authors Quoted, And A entire Index

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3. We define the tangent vector :i; to the path x(t) at the point Xo = :/:(0) to be the equivalence class of all paths y(t) such that y(O) = Xo and such that y(t) is tangent to x(t) at t = O. 4. A tangent vector represented as an equivalence class of paths thmugh a point on the manifold. At any point XQ in a manifold, we consider all smooth paths passing through XQ at time t = O. The property of tangency between two such paths defines an equivalence relation between the paths. The tangent vectors to the manifold at the point XQ are formally defined as the equivalence classes of this relation.

44 2. 3. Secant vectors to a surface. In the limit, as the displacement between points becomes infinitesimal, the secant vectors converge to a tangent vector. at t = 0, say. 27) The paths x(t) and' y(t) are said to be smooth if their coordinates are differentiable functions of the time coordinate t. Henceforth, we shall restrict attention to smooth paths. 28) dt dt for all j = 1, ... , p. It is important to note that although the condition of tangency is expressed in terms of the coordinate system, the tangency property is independent of the choice of coordinates.

2. Thus every smooth path through xodefines a tangent vector at xo. 24) Tangent Vectors and Tangent Spaces Henceforth, we shall assume that MP is a differential manifold. 2 Differential Geometry 2. 3. Secant vectors to a surface. In the limit, as the displacement between points becomes infinitesimal, the secant vectors converge to a tangent vector. at t = 0, say. 26) and Xo = (X01, X02, ... 27) The paths x(t) and' y(t) are said to be smooth if their coordinates are differentiable functions of the time coordinate t.