milano24ore http://www.milano24ore.it/fiera/calendario

argio - Mgglatz

Le téléchargement nécessite un accès à la bibliothèque YouScribe
Tout savoir sur nos offres

29 pages

English

Le téléchargement nécessite un accès à la bibliothèque YouScribe
Tout savoir sur nos offres

A propos
Informations
Extrait

Description

exposé - matière potentielle : womens
exposé

milano24ore 26/01/2012 - 29/01/2012 MACEF International Home Show - Fiera Milano Rho - Hotel a Milano dal 26/01 al 29/01/2012 26/01/2012 - 29/01/2012 CHIBI and CART International Show for Gifts, Perfumery, Jewellery, Paper Products - Fiera Milano Rho - Hotel a Milano dal 26/01 al 29/01/2012 07/02/2012 - 09/02/2012 MILANO UNICA The Italian Textile Exhibition - IDEABIELLA - IDEACOMO - MODA IN - SHIRT AVENUE - quartiere fieramilanocity - Hotel a Milano dal

milano
services for automatic vending
global beauty event
fiera milano
micam shoevent fiera milano calendario
international exhibition
services

Sujets

Exposé

Complemento

Mobile

International

Informations

Publié par	argio
Nombre de lectures	19
Langue	English

Extrait

NASA/TM—2002-211716
An Introduction to Tensors for Students
of Physics and Engineering
Joseph C. Kolecki
Glenn Research Center, Cleveland, Ohio
September 2002The NASA STI Program Office . . . in Profile
Since its founding, NASA has been dedicated to • CONFERENCE PUBLICATION. Collected
the advancement of aeronautics and space papers from scientific and technical
science. The NASA Scientific and Technical conferences, symposia, seminars, or other
Information (STI) Program Office plays a key part meetings sponsored or cosponsored by
in helping NASA maintain this important role. NASA.
The NASA STI Program Office is operated by • SPECIAL PUBLICATION. Scientific,
Langley Research Center, the Lead Center for technical, or historical information from
NASA’s scientific and technical information. The NASA programs, projects, and missions, STI Program Office provides access to the often concerned with subjects having
NASA STI Database, the largest collection of substantial public interest.
aeronautical and space science STI in the world.
The Program Office is also NASA’s institutional • TECHNICAL TRANSLATION. English-
mechanism for disseminating the results of its language translations of foreign scientific
research and development activities. These results and technical material pertinent to NASA’s
are published by NASA in the NASA STI Report mission.
Series, which includes the following report types:
Specialized services that complement the STI
• TECHNICAL PUBLICATION. Reports of Program Office’s diverse offerings include
completed research or a major significant creating custom thesauri, building customized
phase of research that present the results of databases, organizing and publishing research
NASA programs and include extensive data results . . . even providing videos.
or theoretical analysis. Includes compilations
of significant scientific and technical data and For more information about the NASA STI
information deemed to be of continuing Program Office, see the following:
reference value. NASA’s counterpart of peer-
reviewed formal professional papers but • Access the NASA STI Program Home Page
has less stringent limitations on manuscript at http://www.sti.nasa.gov
length and extent of graphic presentations.
• E-mail your question via the Internet to
• TECHNICAL MEMORANDUM. Scientific help@sti.nasa.gov
and technical findings that are preliminary or
of specialized interest, e.g., quick release • Fax your question to the NASA Access
reports, working papers, and bibliographies Help Desk at 301–621–0134
that contain minimal annotation. Does not
contain extensive analysis. • Telephone the NASA Access Help Desk at
301–621–0390
• CONTRACTOR REPORT. Scientific and
technical findings by NASA-sponsored • Write to:
contractors and grantees. NASA Access Help Desk Center for AeroSpace Information
7121 Standard Drive
Hanover, MD 21076NASA/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 2002Available from
NASA Center for Aerospace Information National Technical Information Service
7121 Standard Drive 5285 Port Royal Road
Hanover, MD 21076 Springfield, VA 22100
Available electronically at http://gltrs.grc.nasa.govAn 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 1 “Just walk three km due north.” The denominate number 3 km now has the required additional
directional information attached to it. Such numbers are called vectors. Velocity is a vector since
it has a magnitude and a direction; so is momentum. Quite often, a vector is represented by
components. If you were to tell me that to go from here to your house I must walk three blocks
east, two blocks north, and go up three floors, the vector extending from “here” to “your house”
would have three spatial components:
• Three blocks east,
• Two blocks north,
• Three floors up.
Physically, vectors are used to represent locations, velocities, accelerations, flux densities, field
quantities, etc. The defining equations of the gravitational field in classical dynamics (Newton’s
Law of Universal Gravitation), and of the electromagnetic field in classical electrodynamics
(Maxwell’s four equations) are all given in vector form. Since vectors are higher order quantities
than scalars, the physical realities they correspond to are typically more complex than those
represented by scalars.
A Closer Look at Vectors
The action of a vector is equal to the sum of the actions of its components. Thus, in the example
given above, the vector from “here” to “your house” can be represented as
1V = 3 blocks east + 2 blocks north + 3 floors up
Each component of V contains a vector and a scalar part. The scalar and vector components of V
can be represented as follows:
• Scalar: Let a = 3 blocks, b = 2 blocks, and c = 3 floors be the scalar components; and
• Vector: Let i be a unit vector pointing east, j be a unit vector pointing north, and k be a
unit vector pointing up. (N.B.: Unit vectors are non-denominate, have a magnitude of
unity, and are used only to specify a direction.)
Then the total vector, in terms of its scalar components and the unit vectors, can be written as
V = ai + bj + ck.
This notation is standard in all books on physics and engineering. It is also used in books on
introductory mathematics.
Next, let us look at how vectors combine. First of all, we know that numbers may be combined
in various ways to produce new numbers. For example, six is the sum of three and three or the
product of two and three. A similar logic holds for vectors. Vector rules of combination include
vector addition, scalar (dot or inner) multiplication, and (in three dimensions) cross
multiplication. Two vectors, U and V, can be added to produce a new vector W:
W = U + V.

1
The appropriate symbol to use here is “⇒” rather than “=” since the ‘equation’ is not a strict vector
identity. However, for the sake of clarity, the “⇒” notation has been suppressed both here and later on,
and “=” signs have been used throughout. There is no essential loss in rigor, and the meaning should be
clear to all readers.
NASA/TM2002-211716 2 Vector addition is often pictorially represented by the so-called parallelogram rule. This rule is a
pencil and straightedge construction that is strictly applicable only for vectors in E