Tallarida, R.."Section XIl- Mathematics, Symbols, and Physical C The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Tallarida, R.J. “Section XII – Mathematics, Symbols, and Physical Constants” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
The mathematical equation used to generate this three-dimensional figure is worth thousand words. It represents a single-solitron surface for the sine-Gordon equation w sin w. Among the areas in which the sine-Gordon equation arises is that of wave propagation on nonlinear transmission lines and in semi-conductors. The equation is famous because it is known to have a nonlinear superposition principle obtainable by means of a Backlund Transformation. The sine-Gordon equation is an example of an evolution equation which has an infinite sequence of non-trivial conservation laws so important in the fields of engineering and physics. For further information on the Backlund Transformation see Back lund Transformations and their Application, Rogers and Shadwick, Academic Press, 1982. This three-dimensional projection was generated using the MAPlEs software package MAPLES is one of three important mathematical computer packages that offer a variety of analytical and numerical software for use by scientists, engineers, and mathematicians. This figure was developed by W.K. Schief and C. Rogers and the Center for Dynamical Systems and Nonlinear Studies at Georgia Institute of Technology and the University of New South Wales in Sydney, Australia (Figure courtesy of Schief and Rogers. c 2000 by CRC Press LLC
The mathematical equation used to generate this three-dimensional figure is worth a thousand words. It represents a single-solitron surface for the sine-Gordon equation wuv = sin w. Among the areas in which the sine-Gordon equation arises is that of wave propagation on nonlinear transmission lines and in semi-conductors. The equation is famous because it is known to have a nonlinear superposition principle obtainable by means of a Bäcklund Transformation. The sine-Gordon equation is an example of an evolution equation which has an infinite sequence of non-trivial conservation laws so important in the fields of engineering and physics. For further information on the Bäcklund Transformation see Bäcklund Transformations and their Application, Rogers and Shadwick, Academic Press, 1982. This three-dimensional projection was generated using the MAPLE® software package. MAPLE® is one of three important mathematical computer packages that offer a variety of analytical and numerical software for use by scientists, engineers, and mathematicians. This figure was developed by W.K. Schief and C. Rogers and the Center for Dynamical Systems and Nonlinear Studies at Georgia Institute of Technology and the University of New South Wales in Sydney, Australia. (Figure courtesy of Schief and Rogers.) © 2000 by CRC Press LLC
Mathematics, Symbols and Physical Constants International System of Units(SI) Definitions of SI Base Units Names and Symbols for the SI Base Units SI Derived Units with Special Names and Symbols. Units in Use Together with the SI Conversion Constants and Multipliers Recommended Decimal Multiples and Submultiples.Conversion Factors-Metric to English Conversion Factors-English to Metric .Conversion Factors--General. Temperature Factors Conversion of Temperatures Physical Constants Genera·π Constants· Constants Involving e: Numerical Constants ols and Terminology for Physical and Chemical Quantities Classical Mechanics. Electricity and Magnetism. Electromagnetic Radiation. Solid State Credits Ronald tallarida Temple university T HE GREAT ACHIEVEMENTS in engineering deeply affect the lives of all of us and also serve to remind us of the importance of mathematics. Interest in mathematics has grown steadily with these engineering achievements and with concomitant advances in pure physical science. Whereas scholars in nonscien tific fields, and even in such fields as botany, medicine, geology, etc, can communicate most of the problems and results in nonmathematical language, this is virtually impossible in present-day engineering and physics Yet it is interesting to note that until the beginning of the twentieth century engineers regarded calculus as omething of a mystery. Modern students of engineering now study calculus, as well as differential equations, omplex variables, vector analysis, orthogonal functions, and a variety of other topics in applied analysis. The study of systems has ushered in matrix algebra and, indeed, most engineering students now take linear algebra as a core topic early in their mathematical education This section contains concise summaries of relevant topics in applied engineering mathematics and certain key formulas, that is, those formulas that are most often needed in the formulation and solution of engineering problems. Whereas even inexpensive electronic calculators contain tabular material(e.g, tables of trigonometric and logarithmic functions) that used to be needed in this kind of handbook, most calculators do not give symbolic results. Hence, we have included formulas along with brief summaries that guide their use. In many cases we have added numerical examples, as in the discussions of matrices, their inverses, and their use in the solutions of linear systems. a table of derivatives is included, as well as key applications of the derivative in the solution of problems in maxima and minima, related rates, analysis of curvature, and finding approximate c 2000 by CRC Press LLC
© 2000 by CRC Press LLC XII Mathematics, Symbols, and Physical Constants Greek Alphabet International System of Units (SI) Definitions of SI Base Units • Names and Symbols for the SI Base Units • SI Derived Units with Special Names and Symbols • Units in Use Together with the SI Conversion Constants and Multipliers Recommended Decimal Multiples and Submultiples • Conversion Factors—Metric to English • Conversion Factors—English to Metric • Conversion Factors—General • Temperature Factors • Conversion of Temperatures Physical Constants General • p Constants • Constants Involving e • Numerical Constants Symbols and Terminology for Physical and Chemical Quantities Classical Mechanics • Electricity and Magnetism • Electromagnetic Radiation • Solid State Credits Ronald J. Tallarida Temple University HE GREAT ACHIEVEMENTS in engineering deeply affect the lives of all of us and also serve to remind us of the importance of mathematics. Interest in mathematics has grown steadily with these engineering achievements and with concomitant advances in pure physical science. Whereas scholars in nonscientific fields, and even in such fields as botany, medicine, geology, etc., can communicate most of the problems and results in nonmathematical language, this is virtually impossible in present-day engineering and physics. Yet it is interesting to note that until the beginning of the twentieth century engineers regarded calculus as something of a mystery. Modern students of engineering now study calculus, as well as differential equations, complex variables, vector analysis, orthogonal functions, and a variety of other topics in applied analysis. The study of systems has ushered in matrix algebra and, indeed, most engineering students now take linear algebra as a core topic early in their mathematical education. This section contains concise summaries of relevant topics in applied engineering mathematics and certain key formulas, that is, those formulas that are most often needed in the formulation and solution of engineering problems.Whereas even inexpensive electronic calculators contain tabular material (e.g., tables of trigonometric and logarithmic functions) that used to be needed in this kind of handbook, most calculators do not give symbolic results. Hence, we have included formulas along with brief summaries that guide their use. In many cases we have added numerical examples, as in the discussions of matrices, their inverses, and their use in the solutions of linear systems. A table of derivatives is included, as well as key applications of the derivative in the solution of problems in maxima and minima, related rates, analysis of curvature, and finding approximate T
by numerical methods. A list of infinite series, along with the interval of convergence of each, is also the two branches of calculus, integral calculus is richer in its applications, as well as in its theoretical content. Though the theory is not emphasized here, important applications such as finding areas, lengths volumes, centroids, and the work done by a nonconstant force are included. Both cylindrical and spherical olar coordinates are discussed, and a table of integrals is included. Vector analysis is summarized in a separate section and includes a summary of the algebraic formulas involving dot and cross multiplication, frequently needed in the study of fields, as well as the important theorems of Stokes and Gauss. The part on special functions includes the gamma function, hyperbolic functions, Fourier series, orthogonal functions, and both Laplace and z-transforms. The Laplace transform provides a basis for the solution of differential equations and is fundamental to all concepts and definitions underlying analytical tools for describing feedback control systems. The z-transform, not discussed in most applied mathematics books, is most useful in the analysis of discrete signals as, for example, when a computer receives data sampled at some prespecified time interval. The Bessel functions, also called cylindrical functions, arise in many physical applications, such as the heat transfer in a"long"cylinder, whereas the other orthogonal functions discussed--Legendre, Hermite, and Laguerre olynomials-are needed in quantum mechanics and many other subjects(e.g, solid-state electronics)that use concepts of modern physics. The world of mathematics, even applied mathematics, is vast. Even the best mathematicians cannot keep up with more than a small piece of this world. The topics included in this section, however, have withstood the test of time and, thus, are truly core for the modern engineer. This section also incorporates tables of physical constants and symbols widely used by engineers. While not exhaustive, the constants, conversion factors, and symbols provided will enable the reader to accommodate a majority of the needs that arise in design, test, and manufacturing functions e 2000 by CRC Press LLC
© 2000 by CRC Press LLC roots by numerical methods. A list of infinite series, along with the interval of convergence of each, is also included. Of the two branches of calculus, integral calculus is richer in its applications, as well as in its theoretical content. Though the theory is not emphasized here, important applications such as finding areas, lengths, volumes, centroids, and the work done by a nonconstant force are included. Both cylindrical and spherical polar coordinates are discussed, and a table of integrals is included. Vector analysis is summarized in a separate section and includes a summary of the algebraic formulas involving dot and cross multiplication, frequently needed in the study of fields, as well as the important theorems of Stokes and Gauss. The part on special functions includes the gamma function, hyperbolic functions, Fourier series, orthogonal functions, and both Laplace and z-transforms. The Laplace transform provides a basis for the solution of differential equations and is fundamental to all concepts and definitions underlying analytical tools for describing feedback control systems. The z-transform, not discussed in most applied mathematics books, is most useful in the analysis of discrete signals as, for example, when a computer receives data sampled at some prespecified time interval. The Bessel functions, also called cylindrical functions, arise in many physical applications, such as the heat transfer in a “long” cylinder, whereas the other orthogonal functions discussed—Legendre, Hermite, and Laguerre polynomials—are needed in quantum mechanics and many other subjects (e.g., solid-state electronics) that use concepts of modern physics. The world of mathematics, even applied mathematics, is vast. Even the best mathematicians cannot keep up with more than a small piece of this world. The topics included in this section, however, have withstood the test of time and, thus, are truly core for the modern engineer. This section also incorporates tables of physical constants and symbols widely used by engineers. While not exhaustive, the constants, conversion factors, and symbols provided will enable the reader to accommodate a majority of the needs that arise in design, test, and manufacturing functions
Mathematics ymbois, an Physical Constants Greek Alphabet Greek English ter name equivalent ame N ABr△EZ Gamma oP Omicron P Kappa Lambda TYΦxYΩ International System of Units (Si) The International System of units(Si)was adopted by the 1lth General Conference on Weights and Measures (CGPM) in 1960. It is a coherent system of units built form seven SI base units, one for each of the seven dimensionally independent base quantities: they are the meter, kilogram, second, ampere, kelvin, mole, and candela, for the dimensions length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity, respectively. The definitions of the SI base units are given below. The SI derived units are expressed as products of powers of the base units, analogous to the corresponding relations between physical quantities but with numerical factors equal to unity. In the International System there is only one SI unit for each physical quantity. This is either the appropriate SI base unit itself or the appropriate SI derived unit. However, any of the approved decimal prefixes, called SI prefixes, may be used to construct decimal multiples or submultiples of Si units It is recommended that only Si units be used in science and technology(with SI prefixes where appropriate) Where there are special reasons for making an exception to this rule, it is recommended always to define the units used in terms of SI units. This section is based on information supplied by IUPAC. Definitions of si Base Units Meter--The meter is the length of path traveled by light in vacuum during a time interval of 1/299 792 458 d (17th CGPM, 1983) Kilogram--The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram(3rd CGPM, 1901) Second-The second is the duration of9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom(13th CGPM, 1967) Ampere--The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 x 10- newton per meter of length(9th CGPM, 1948) e 2000 by CRC Press LLC
© 2000 by CRC Press LLC Mathematics, Symbols, and Physical Constants Greek Alphabet International System of Units (SI) The International System of units (SI) was adopted by the 11th General Conference on Weights and Measures (CGPM) in 1960. It is a coherent system of units built form seven SI base units, one for each of the seven dimensionally independent base quantities: they are the meter, kilogram, second, ampere, kelvin, mole, and candela, for the dimensions length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity, respectively. The definitions of the SI base units are given below. The SI derived units are expressed as products of powers of the base units, analogous to the corresponding relations between physical quantities but with numerical factors equal to unity. In the International System there is only one SI unit for each physical quantity. This is either the appropriate SI base unit itself or the appropriate SI derived unit. However, any of the approved decimal prefixes, called SI prefixes, may be used to construct decimal multiples or submultiples of SI units. It is recommended that only SI units be used in science and technology (with SI prefixes where appropriate). Where there are special reasons for making an exception to this rule, it is recommended always to define the units used in terms of SI units. This section is based on information supplied by IUPAC. Definitions of SI Base Units Meter—The meter is the length of path traveled by light in vacuum during a time interval of 1/299 792 458 of a second (17th CGPM, 1983). Kilogram—The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram (3rd CGPM, 1901). Second—The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom (13th CGPM, 1967). Ampere—The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 ¥ 10–7 newton per meter of length (9th CGPM, 1948). Greek Greek English Greek Greek English letter name equivalent letter name equivalent A a Alpha a N n Nu n B b Beta b X x Xi x G g Gamma g O o Omicron o˘ D d Delta d P p Pi p E e Epsilon e˘ R r Rho r Z z Zeta z S s Sigma s H h Eta e – T t Tau t Q qJ Theta th U u Upsilon u I i Iota i F fj Phi ph K k Kappa k C c Chi ch L l Lambda l Y y Psi ps M m Mu m W w Omega o–