Multidimensional Analysis

Algebras and Systems for Science and Engineering
1995
ISBN: 978-0-387-94417-3
Verlag: Springer

Buch, Englisch, 236 Seiten, Format (B × H): 160 mm x 241 mm, Gewicht: 1180 g

This book deals with the mathematical properties of dimensioned quantities, such as length, mass, voltage, and viscosity.
Beginning with a careful examination of how one expresses the numerical results of a measurement and uses these results in subsequent manipulations, the author rigorously constructs the notion of dimensioned numbers and discusses their algebraic structure. The result is a unification of linear algebra and traditional dimensional analysis that can be extended from the scalars to which the traditional analysis is perforce restricted to multidimensional vectors of the sort frequently encountered in engineering, systems theory, economics, and other applications.

Zielgruppe

Research

Autoren/Hrsg.

Hart, George W.

Fachgebiete

• Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure
• Mathematik | Informatik Mathematik Mathematische Analysis Moderne Anwendungen der Analysis
• Naturwissenschaften Physik Physik Allgemein Geschichte der Physik
• Mathematik | Informatik EDV | Informatik Professionelle Anwendung Computer-Aided Design (CAD)
• Naturwissenschaften Physik Physik Allgemein Experimentalphysik
• Mathematik | Informatik EDV | Informatik Angewandte Informatik Computeranwendungen in Wissenschaft & Technologie
• Technische Wissenschaften Technik Allgemein Computeranwendungen in der Technik
• Naturwissenschaften Biowissenschaften Angewandte Biologie Biomathematik
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik

Inhaltsverzeichnis

0. Introductory.- 0.1 Physical Dimensions.- 0.2 Mathematical Dimensions.- 0.3 Overview.- Exercises.- 1. The Mathematical Foundations of Science and Engineering.- 1.1 The Inadequacy of Real Numbers.- 1.2 The Mathematics of Dimensioned Quantities.- 1.3 Conclusions.- Exercises.- 2. Dimensioned Linear Algebra.- 2.1 Vector Spaces and Linear Transformations.- 2.2 Terminology and Dimensional Inversion.- 2.3 Dimensioned Scalars.- 2.4 Dimensioned Vectors.- 2.5 Dimensioned Matrices.- Exercises.- 3. The Theory of Dimensioned Matrices.- 3.1 The Dimensional Freedom of Multipliable Matrices.- 3.2 Endomorphic Matrices and the Matrix Exponential.- 3.3 Square Matrices, Inverses, and the Determinant.- 3.4 Squarable Matrices and Eigenstructure.- 3.5 Dimensionally Symmetric Multipliable Matrices.- 3.6 Dimensionally Hankel and Toeplitz Matrices.- 3.7 Uniform, Half Uniform, and Dimensionless Matrices.- 3.8 Conclusions.- 3.A Appendix: The n + m ? 1 Theorem.- Exercises.- 4. Norms, Adjoints, and Singular Value Decomposition.- 4.1 Norms for Dimensioned Spaces.- 4.2 Dimensioned Singular Value Decomposition (DSVD).- 4.3 Adjoints.- 4.4 Norms for Nonuniform Matrices.- 4.5 A Control Application.- 4.6 Factorization of Symmetric Matrices.- Exercises.- 5. Aspects of the Theory of Systems.- 5.1 Differential and Difference Equations.- 5.2 State-Space Forms.- 5.3 Canonical Forms.- 5.4 Transfer Functions and Impulse Responses.- 5.5 Duals and Adjoints.- 5.6 Stability.- 5.7 Controllability, Observability, and Grammians.- 5.8 Expectations and Probability Densities.- Exercises.- 6. Multidimensional Computational Methods.- 6.1 Computers and Engineering.- 6.2 Representing and Manipulating Dimensioned Scalars.- 6.3 Dimensioned Vectors.- 6.4 Representing Dimensioned Matrices.- 6.5 Operations on DimensionedMatrices.- 6.6 Conclusions.- Exercises.- 7. Forms of Multidimensional Relationships.- 7.1 Goals.- 7.2 Operations.- 7.3 Procedure.- Exercises.- 8. Concluding Remarks.- 9. Solutions to Odd-Numbered Exercises.- References.