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      Introduction

      Euclidean Geometry was one of the most important branches of mathematics for the last 2000 years. It was also the main tool used by scientists for the development of astronomy. In 300 B.C., Euclid treated geometry as a deductive system. In 1621, Sir Henry Savile raised some questions concerning what he called “two blemishes” in geometry: the theory of proportion and the theory of parallels. Euclid’s axiom of parallels (postulate V in the first book of Elements) propagates that any two given lines in a plane, when produced indefinitely, will intersect if the sum of two interior angles made by a transversal with these lines is less than two right angles. In 1826, a Russian mathematician, Nicolai Lobachevski, presented a paper to the mathematician faculty of the University of Kazan based on the assumption that it is possible to draw through any point in a plane two lines parallel to a given line. Hungarian mathematician John Bolyai published some results in 1831, which, conceptually, have little difference from those of Lobachevski and perhaps it contains a deeper appreciation of the metric properties of space. However, it was only after Riemann’s profound dissertation on the hypotheses that the underlying foundations of geometry appeared posthumously in 1867, showing the importance of the metric concepts in geometry.

      Riemann appeared to have been unaware of the work of Lobachevski and Bolyai, although it was well known to Gauss. Later, Beltrami published his classical paper on the interpretation of non-Euclidean geometries (1868) in which he analyzed the work of Lobachevski, Bolyai, and Riemann and stressed the fact that the metric properties of space are mere definitions. It appears that three consistent geometries are possible on surfaces of constant curvatures: Lobachevskian on a surface of constant negative curvature, Riemannian on a surface of constant positive curvature, and Euclidean on a surface of zero curvature. These geometries are called hyperbolic, elliptic, and parabolic, respectively.

      Tensor calculus is concerned with the study of abstract objects, called tensors, which are independent of frames of reference used to describe them. In the early part of the 17^th century, geometry was developed by French mathematician Descartes. The great physicist W. Voigi discovered tensors and gave them this name in his remarkable book A Textbook of Crystal Physics published in 1910. Ricci and his student, Levi-cita, (1901) have been developing the subject of tensors, but it is well known that Einstein’s use of tensors as a tool in his general theory of relativity (1914) was mainly responsible for the sudden emergence of tensor calculus as a popular field of mathematical activities. The application of tensors in the field of mathematics and physics was mainly accelerated after the publication of Einstein’s famous paper The General Theory of Relativity.

      Tensor is a generalization of the term vector and Tensor Calculus is a generalization of vector analysis. The concept of invariance of mathematical objects, under coordinate transformations, permeates the structure of tensor analysis to such an extent that it is important to get at the outset a clear notion of the particular brand of invariance we have in mind. In the given reference frame, a point P is determined by a set of coordinates, xi. If the coordinate system is changed, point P is described by a new set of coordinates, but the transformation of the coordinates does nothing to the point itself.

      Here, we discuss how the term tensor may be considered as a generalization of the term vector.

      We start with a two-dimensional Euclidean space, E₂, provided with a system of rectangular Cartesian coordinates. Let P and Q be two points of E₂, with (x₁, x₂) and (y₁, y₂) as their respective coordinates. Then, the coordinates of the directed line segment

      Denoting x₁ − y₁ and x₂ – y₂ by z₁ and z₂ respectively, the coordinates of

      (0.1)

      Next, we consider an orthogonal transformation of coordinate axes given by

      (0.2)

      where

We can express (0.2) as follows:

(0.3)

denoted by (x₁, x₂) and (y₁, y₂), the coordinates of Q and P in the new coordinates system. Then, the coordinates of vector

(0.4)