Tensory - english

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NASA/TM—2002-211716
An Introduction to Tensors for Students
of Physics and Engineering
Joseph C. Kolecki
Glenn Research Center, Cleveland, Ohio
September 2002
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NASA/TM—2002-211716
An Introduction to Tensors for Students
of Physics and Engineering
Joseph C. Kolecki
Glenn Research Center, Cleveland, Ohio
National Aeronautics and
Space Administration
Glenn Research Center
September 2002
 Available from
NASA Center for Aerospace Information
7121 Standard Drive
Hanover, MD 21076
National Technical Information Service
5285 Port Royal Road
Springfield, VA 22100
Available electronically at
 An Introduction To Tensors
for Students of Physics and Engineering
Joseph C. Kolecki
National Aeronautics and Space Administration
Glenn Research Center
Cleveland, Ohio 44135
Tensor analysis is the type of subject that can make even the best of students shudder. My own
post-graduate instructor in the subject took away much of the fear by speaking of an implicit
rhythm
in the peculiar notation traditionally used, and helped me to see how this rhythm plays its
way throughout the various formalisms.
Prior to taking that class, I had spent many years “playing” on my own with tensors. I found the
going to be tremendously difficult, but was able, over time, to back out some physical and
geometrical considerations that helped to make the subject a little more transparent. Today, it is
sometimes hard not to think in terms of tensors and their associated concepts.
This article, prompted and greatly enhanced by Marlos Jacob, whom I’ve met only by e-mail, is
an attempt to record those early notions concerning tensors. It is intended to serve as a bridge
from the point where most undergraduate students “leave off” in their studies of mathematics to
the place where most texts on tensor analysis begin. A basic knowledge of vectors, matrices, and
physics is assumed. A semi-intuitive approach to those notions underlying tensor analysis is
given via scalars, vectors, dyads, triads, and similar higher-order vector products. The reader
must be prepared to do some mathematics and to think.
For those students who wish to go beyond this humble start, I can only recommend my
professor’s wisdom: find the
rhythm
in the mathematics and you will fare pretty well.
Beginnings
At the heart of all mathematics are numbers.
If I were to ask how many marbles you had in a bag, you might answer, “Three.” I would find
your answer perfectly satisfactory. The ‘bare’ number 3, a magnitude, is sufficient to provide the
information I seek.
If I were to ask, “How far is it to your house?” and you answered, “Three,” however, I would
look at you quizzically and ask, “Three
what
?” Evidently, for this question, more information is
required. The bare number 3 is no longer sufficient; I require a ‘
denominate
’ number – a number
with a name.
Suppose you rejoindered, “Three km.” The number 3 is now
named
as representing a certain
number of
km
. Such numbers are sometimes called
scalars
. Temperature is represented by a
scalar. The total energy of a thermodynamic system is also represented by a scalar.
If I were next to ask “Then how do I get to your house from here?” and you said, “Just walk
three km,” again I would look at you quizzically. This time, not even a denominate number is
sufficient; it is necessary to specify a distance or magnitude, yes, but in which direction?
NASA/TM
2002-211716
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