Tensor and Vector Analysis

With Applications to Differential Geometry

C. E. Springer

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486320915
• Topic: Matematica
• Subtopic: Analisi vettoriale

Concise and user-friendly, this college-level text assumes only a knowledge of basic calculus in its elementary and gradual development of tensor theory. The introductory approach bridges the gap between mere manipulation and a genuine understanding of an important aspect of both pure and applied mathematics.
Beginning with a consideration of coordinate transformations and mappings, the treatment examines loci in three-space, transformation of coordinates in space and differentiation, tensor algebra and analysis, and vector analysis and algebra. Additional topics include differentiation of vectors and tensors, scalar and vector fields, and integration of vectors. The concluding chapter employs tensor theory to develop the differential equations of geodesics on a surface in several different ways to illustrate further differential geometry.

1

Coordinate Transformations and Mappings

1–1. Two Aspects. The reader will recall two transformations from analytical geometry in the plane. The first is the translation which can be represented by the equations

and the second is the rotation described by the equations

Equations (1) may be interpreted in two ways. In the first way a point P has cartesian coordinates (x,y) referred to the x- and y-axes through the origin O, and the same point P also has coordinates

referred to a set of

- and

-axes parallel to the x- and y-axes and passing through the point Ō. Notice that the origin O has coordinates (h,k) referred to the “barred” axes, while the point Ō has coordinates (− h, − k) in the xy system. In this first aspect of equations (1), any point P remains fixed but its identification or “name” changes when the frame of reference is changed, and equations (1) serve as a sort of dictionary which reveals the “name”

if the “name” (x,y) is known. One speaks, therefore, of a transformation of coordinates effected by equations (1), which is the result of a translation of the coordinate axes.

A second interpretation of equations (1) is the following. There is only one set of coordinate axes through a fixed origin O. The point P(x,y) is mapped or transported into a different position P with coordinates

referred to the single set of xy axes through O. One speaks of equations (1) as representing a mapping of the points of the plane by a translation determined by the parameters h and k. Notice that if both h and k are not zero, then no point in the finite plane can remain fixed under the mapping defined by equations (1). In projective geometry, the mapping of (1) is styled an affine mapping because finite points map into finite points and so-called points at infinity map into points at infinity. Observe that h and k are parameters, each of which could be assigned an arbitrary numerical value. Hence, one could say that equations (1) represent a doubly infinite set of mappings.

The two aspects discussed here have been referred to as the “alias” and “alibi” aspects. Under the “alias” connotation points ...