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Tensor Elements
We attribute tensor calculus to the Italian mathematicians Ricci and Levi-Civita,4 who provided the modeling language for Einstein to formulate his famous General Theory of Relativity.5 More recently, tensor calculus is also penetrating the applied and engineering sciences. Some of the references that shaped my research are the three volumes by Duschek and Hochrainer,6 which emphasize the coordinate invariancy of physical quantities; the book by Wrede,7 with its concept of the rotational time derivative; and the engineering text by Betten.8
The world of the engineer is simple, as long as he remains in the solar system and travels at a fraction of the speed of light. His space is Euclidean and has three
dimensions. Newtonian mechanics is adequate to describe the dynamic phenomena. In flight mechanics we can even further simplify the Euclidean metric to finite differences Л, the so-called Cartesian metric
з
As2 — Ax2 + Ax + Ax2 = ^2 &хї
/=1
The elements Дх, are mutually orthogonal, and the metric expresses the Pythagorean theorem of how to calculate the finite distance As. In this world tensors are called Cartesian tensors. As we will see, they are particularly simple to use and completely adequate for our modeling tasks.
The elements of Cartesian tensor calculus are few. I will summarize them for you, discuss products of tensors, and wrap it up with some examples. Keep an open mind! I will break with some traditional concepts of vector mechanics in favor of a modem treatment of modeling of aerospace vehicles. Before we discuss Cartesian tensors however, we need to define coordinates and coordinate systems.
