NASA/TP—2005-213115 Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity Joseph C. Kolecki Glenn Research Center, Cleveland, Ohio April 2005 The 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, NASA 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 [email protected] 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 NASA Center for AeroSpace Information 7121 Standard Drive Hanover, MD 21076 NASA/TP—2005-213115 Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity Joseph C. Kolecki Glenn Research Center, Cleveland, Ohio National Aeronautics and Space Administration Glenn Research Center April 2005 Acknowledgments To Dr. Ken DeWitt of Toledo University, I extend a special thanks for being a guiding light to me in much of my advanced mathematics, especially in tensor analysis. Years ago, he made the statement that in working with tensors, one must learn to find—and feel—the rhythm inherent in the indices. He certainly felt that rhythm, and his ability to do so made a major difference in his approach to teaching the material and enabling his students to comprehend it. He read this work and made many valuable suggestions and alterations that greatly strengthened it. I wish to also recognize Dr. Harold Kautz’s contribution to the section Magnetic Permeability and Material Stress, which was derived from a conversation with him. Dr. Kautz has been my colleague and part-time mentor since 1973. Available 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.gov Contents Summary ............................................................................................................................................ 1 Introduction ........................................................................................................................................ 1 Alegbra ............................................................................................................................................... 1 Statement of Core Ideas ............................................................................................................... 1 Number Systems .......................................................................................................................... 2 Numbers, Denominate Numbers, and Vectors............................................................................. 3 Formal Presentation of Vectors.................................................................................................... 3 Vector Arithmetic ........................................................................................................................ 5 Dyads and Other Higher Order Products ..................................................................................... 8 Dyad Arithmetic........................................................................................................................... 10 Components, Rank, and Dimensionality...................................................................................... 13 Dyads as Matrices ........................................................................................................................ 14 Fields............................................................................................................................................ 15 Magnetic Permeability and Material Stress ................................................................................. 16 Location and Measurement: Coordinate Systems........................................................................ 18 Multiple Coordinate Systems: Coordinate Transformations........................................................ 19 Coordinate Independence............................................................................................................. 20 Coordinate Independence: Another Point of View ...................................................................... 21 Coordinate Independence of Physical Quantities: Some Examples............................................. 23 Metric or Fundamental Tensor..................................................................................................... 24 Coordinate Systems, Base Vectors, Covariance, and Contravariance ......................................... 27 Kronecker’s Delta and the Identity Matrix .................................................................................. 29 Dyad Components: Covariant, Contravariant, and Mixed........................................................... 30 Relationship Between Covariant and Contravariant Components of a Vector............................ 30 st s Relation Between gij, g , and δw ................................................................................................. 32 Inner Product as an Operation Involving Mixed Indices ............................................................. 32 General Mixed Component: Raising and Lowering Indices........................................................ 34 Tensors: Formal Definitions ........................................................................................................ 35 Is the Position Vector a Tensor? .................................................................................................. 38 The Equivalence of Coordinate Independence With the Formal Definition for a Rank 1 Tensor (Vector)......................................................................................................... 39 Coordinate Transformation of the Fundamental Tensor and Kronecker’s Delta......................... 40 Two Examples From Solid Analytical Geometry........................................................................ 40 Calculus .............................................................................................................................................. 42 Statement of Core Idea................................................................................................................. 42 First Steps Toward a Tensor Calculus: An Example From Classical Mechanics........................ 42 Base Vector Differentials: Toward a General Formulation ......................................................... 48 Another Example From Polar Coordinates.................................................................................. 50 Base Vector Differentials in the General Case ............................................................................ 51 Tensor Differentiation: Absolute and Covariant Derivatives ...................................................... 55 k Tensor Character of Γwt .............................................................................................................. 56 Differentials of Higher Rank Tensors.......................................................................................... 58 Product