ÆTHERFORCE ENGfflEEMG MATHEMATICS A SERIES OF LECTURES DELIVERED AT UNION COLLEGE BY CHARLES PROTEUS STEIMET2, A.M., Pn.D. PAST TKKSIDFNT UIFIMC' V\ INSTITUTE OF LLLCTRICVL LXGINLERS McGRAW-HILL BOOK COMPANY 239 WEST 39TH STREET, NEW YORK 6 BOUVERIE STEEET, LONDON, E.G. 1911 ÆTHERFORCE Copyright, JO 11, BV McGRAW-lIILL BOOK COMPANY ÆTHERFORCE PREFACE. THE following work embodies the subject-matter of a lecture course which I have given to the junior and senior electrical engineering students of Union University for a number of years. It is generally conceded that a fair knowledge of mathe- matics is necessary to the engineer, and especially the electrical engineer. For the latter, however, some branches of mathe- matics are of fundamental importance, as the algebra of the general number, the exponential and trigonometric series, etc., which are seldom adequately treated, and often not taught at all in the usual text-books of mathematics, or in the college course of analytic geometry and calculus given to the engineer- ing students, and, therefore, electrical engineers often possess little knowledge of these subjects. As the result, an electrical engineer, even if he possess a fail' knowledge of mathematics, may often find difficulty in dealing with problems, through lack of familiarity with these branches of mathematics, which have become of importance in electrical engineering, and may also find difficulty in looking up information on these subjects. In the same way the college student, when beginning the study of electrical engineering theory, after completing his general course of mathematics, frequently finds' himself sadly deficient in the knowledge of mathematical subjects, of which a complete familiarity is required for effective understanding of electrical engineering theory. It was this experience which led me some years ago to start the course of lectures which is reproduced in the following pages. I have thus attempted to bring together and discuss explicitly, with numerous practical applications, all those branches of mathematics which are of special importance to the electrical engineer. Added thereto ÆTHERFORCE vi PKEIWE. are a number of subjects which experience has shown me to be important for the effective and expeditious execution of electrical engineering calculations. Merc theoretical knowledge of mathematics is not sufficient for the engineer, but it must be accompanied by ability to apply it and derive resultsto carry out numerical calculations. It is not sufficient to know how a phenomenon occurs, and how it may be calculated, but very often there is a wide gap between this knowledge and the ability to carry out the calculation; indeed, frequently an attempt to apply the theoretical knowledge to derive numerical results leads, even in simple problems, to apparently hopeless complication and almost endless calculation, so that all hope of getting reliable results vanishes. Thus considerable space has been devoted to the discussion of methods of calculation, the use of curves and their evaluation, and other kindred subjects requisite for effective engineering work, Thus the following work is not intended as a complete course in mathematics, but as supplementary to the general college course of mathematics, or to the general knowledge of mathematics which every engineer and really every educated man should possess. In illustrating the mathematical discussion, practical examples, usually taken from the field of electrical engineer- ing, have been given and discussed. These are sufficiently numerous that any example dealing with a phenomenon with which the reader is not yet familiar may be omitted and taken up at a later time. As appendix is given a descriptive outline of the intro- duction to the theory of functions, since the electrical engineer should be familiar with the general relations between the different functions which he meets. In relation to " Theoretical Elements of Electrical Engineer- ing/' "Theory and Calculation of Alternating Current Phe- nomena/ 7 and " Theory and Calculation of Transient Electric Phenomena/' the following work is intended as an introduction and explanation of the mathematical side, and the most efficient method of study, appears to me, to start with " Electrical Engineering Mathematics," and after entering its third chapter, to take up the reading of the first section of " Theo- retical Elements," and then parallel the study of " Electrical ÆTHERFORCE PREFACE. vii Engineering Mathematics/' " Theoretical Elements of Electrical Engineering/' and " Theory and Calculation of Alternating Current Phenomena/' together with selected chapters from "Theory and Calculation of Transient Electric Phenomena/' and after this, once more systematically go through all four books. CHARLES P. STEINMETZ. SCHENECTADY, N. Y., December, 1910, ÆTHERFORCE ÆTHERFORCE CONTENTS. PAGE PREFACE v CHAPTER I. THE GENERAL NUMBER. A. THE SYSTEM OF NUMBERS. 1. Addition and Subtraction. Origin of numbers. Counting and measuring. Addition. Subtraction as reverse operation of addition 1 2. Limitation of subtraction. Subdivision of the absolute numbers into positive and negative 2 3. Negative number a mathematical conception like the imaginary number. Cases where the negative number has a physical meaning, and cases where it has not 4 4. Multiplication and Division. Multiplication as multiple addi- tion, Division as its reverse operation. Limitation of divi- sion 6 5. The fraction as mathematical conception. Cases where it has a physical meaning, and cases where it has not 8 C. Involution and Evolution. Involution as multiple multiplica- tion. Evolution as its reverse operation. Negative expo- nents 9 7. Multiple involution leads to no new operation 10 8. Fractional exponents 10 9. Irrational Numbers. Limitation of evolution. Endless decimal fraction. Rationality of the irrational number 11 10. Quadrature numbers. Multiple values of roots. Square root of negative quantity representing quadrature number, or rota- tion by 90 13 H. Comparison of positive, negative and quadrature numbers. Reality of quadrature number. Cases where it has a physical meaning, and cases where it has not 14 12. General Numbers. Representation of the plane by the general number. Its relation to rectangular coordinates 16 13. Limitation of algebra by the general number. Roots of the unit. Number of such roots, and their relation 18 14. The two reverse operations of involution 19 ix ÆTHERFORCE x CONTENTS. PAGE 15. Logarithmation. Relation between logarithm and exponent of involution. Reduction to other base. Logarithm of negative quantity 20 16. Quaternions. Vector calculus of space 22 17. Space rotors and their relation. Super algebraic nature of space analysis , 22 B. ALGEBRA OF THE GENERAL NUMBER OF COMPLEX QUANTITY. Rectangular and Polar Coordinates . . 25 IS. Powers of j. Ordinary or real, and quadrature or imaginary number. Relations 25 19. Conception of general number by point of plane in rectangular coordinates; in polar coordinates. Relation between rect- angular and polar form 26 20. Addition and Subtraction. Algebraic and geometrical addition and subtraction. Combination and resolution by parallelo- gram law 28 21. Denotations 30 22. Sign of vector angle. Conjugate and associate numbers. Vec- tor analysis 30 23. Instance of steam path of turbine 33 24. Multiplication. Multiplication in rectangular coordinates. 38 25. Multiplication in polar coordinates. . Vector and operator 38 26. Physical meaning of result of algebraic operation. Representa- tion of result 40 27. Limitation of application of algebraic operations to physical quantities, and of the graphical representation of the result. Graphical representation of algebraic operations between current, voltage and impedance 40 28. Representation of vectors and of operators 42 29. Division. Division in rectangular coordinates 42 30. Division in polar coordinates 43 31. Involution and Evolution. Use of polar coordinates 44 32. Multiple values of the result of evolution. Their location in the plane of the general number. Polyphase and n phase systems of numbers 45 33. The n values of Vl and their relation 46 34. Evolution in rectangular coordinates. Complexity of result 47 35. Reduction of products and fractions of general numbers by polar representation. Instance 48 36. Exponential representations of general numbers. The different forms of the general number 49 37. Instance of use of exponential form in solution of differential equation 50 ÆTHERFORCE CONTENTS. xi PAGE 38. Logarithmation, Resolution of the logarithm of a general number 51 CHAPTER II, THE POTENTIAL SERIES AND EXPONENTIAL FUNCTION. A. GENERAL, 39 The infinite series of powers of a; 52 40. Approximation by series 53 41. Alternate and one-sided approximation 54 42. Convergent and divergent series 55 43. Range of convergency. Several series of different ranges for same expression 56 44 Discussion of convergency in engineering applications , . 57 45. Use of series for approximation of small terms. Instance of electric circuit 58 46. Binomial theorem for development in series. Instance of in- ductive circuit 59 47. Necessity of development in series. Instance of a,rc of hyperbola 60 48. Instance of numerical calculation of log (1 -fa;) 63 B. DIFFERENTIAL EQUATIONS, 49. Character of most differential equations of electrical engineering, Their typical forms 64 dy 50. -j il' Solution by scries, by method of indeterminate co- dx efficients , 65 dz 51. 7- az. Solution by indeterminate coefficients 68 dx 52. Integration constant and terminal conditions 68 53. Involution of solution. Exponential function 70 54. Instance of rise of field current in direct current shunt motor . . 72 55. Evaluation of inductance, and numerical calculation 75 56. Instance of condenser discharge through resistance 76 $y 57. Solution Qt-=ay by indeterminate coefficients, by exponential function , 78 58. Solution by trigonometric functions , . . , 81 59. Relations between trigonometric functions and exponential func- tions with imaginary exponent, and inversely 83 60. Instance of condenser discharge through inductance. The two integration constants and terminal conditions 84 61. Effect of resistance on the discharge. The general differential equation 86 ÆTHERFORCE [...]... Exponential function 284 Hyperbolic functions 285 II ÆTHE ORCE RF ÆTHE ORCE RF ENGINEERING MATHEMATICS CHAPTER I THE GENERAL NUMBER A THE SYSTEM OF NUMBERS Addition and Subtraction i From the operation of counting and measuring arose the and finally, more or less, art of figuring, arithmetic, algebra, the entire structure of mathematics During the development of the human race throughout the which is... Degrees of exactness: magni- 252 tude, approximate, exact 254 of decimals ENGINEERING DATA 164 INTELLIGIBILITY OF Curve plotting for showing shape of function, and for record of numerical valuer 256 165 Scale of curves 259 Principles 260 166 Completeness of record 167 RELIABILITY OF NUMERICAL CALCULATIONS Necessity of 261 reliability in engineering calculations 168 Methods of checking 169 Some Curve plotting... 4, 5, and then count a second bunch 1 i, now put of horses, 2 3; *j, the second bunch together with the bunch, and count them That is, first one, into ono after counting the horses ÆTHE ORCE RF ENGINEERING MATHEMATICS 2 of the first bunch, to count those of the second we continue bunch, thus: 2, 1, 3, 4, 5 -G, 7, 8; which gives addition, 5+3-8; or, in general, a+l>=c We may take away again the second... point, just as in Fig, 2 That IKS, 5-3=2 (Fig 2), 5-7=2 (Fig 3) In the case where we can subtract 7 from distance from the starting point as 5, when we we get the same subtract 3 from 5, ÆTHE ORCE RF ENGINEERING MATHEMATICS 4 AC but the distance the same, 2 stops, in Fig 3, while in Fig 2, is different in character, the one the other toward the right, is toward the as left, That means, we have two kinds... a posnumber, itive unit, but the negative unit, multiplied with itself, does ovor, a difference exists not remain a negative unit, but becomes positive: (-l)X(-l)=(+l),andnot =(-1) ÆTHE ORCE RF ENGINEERING MATHEMATICS 6 northern latitude and going 7 cleg, Starting from 5 deg 2 deg southern latitude, which may bo south, brings us to expresses thus, +5 cleg, latitude -7 deg latitude = -2 clog, latitude... by 5 to make 5 groups; that or, as is we is, 12 horses 12 horses usually say; impossible; divided by 5 gives 2 horses and 2 horses left over, which is written, 12 -r=2, remainder 2 ÆTHE ORCE RF ENGINEERING MATHEMATICS 8 Thus it is seen that the reverse operation of multiplication, or division, cannot always be carried out divide them into 5 If we have 10 apples, and apples in each group, and one apple... it would by definition itself we get, 4 this successive 6 -i-4=4 5 division by 4 5 -r4=4 4 ; 4 is ; carried get the following series: =42 =41 =4 - 42 i = or, in general, ? ; ~ 6= a&' ÆTHE ORCE RF ENGINEERING MATHEMATICS 10 & as a~ Thus, powers with negative exponents; reciprocals of the 7 From same powers with the arc , ~ positive exponents: b the definition of involution then follows, a b Xan =d' +n... integer exponents, as 4 =64, can always can also be carried In carried out cases, evolution 9, be out many For instance, it cannot be carried out while, in other cases, For instance, ÆTHE ORCE RF ENGINEERING MATHEMATICS 12 $, we Attempting to calculate get, $=1.4142135 , and no matter how find, far we carry the calculation; wo never an end, but get an endless decimal fraction; that is, no number exists... positive toward the left, would give the negative toward the right) If then we take a number, as +2, which represents a distance AB t and multiply by (-1), we get the distance AC~ -2 ÆTHE ORCE RF ENGINEERING MATHEMATICS 14 in opposite direction from' A, and multiply by (-1), we tion by (-1) If of we iS=+2; that reverses the direction, turns multiply +2 /:: l we get by \ we take if Inversely, get it through... in In other problems, as when dealing with time, which Fig 6 has only two directions, past and future, the quadrature numbers are not applicable, but only the positive and 'negative ÆTHE ORCE RF ENGINEERING MATHEMATICS 16 In numbers still other problems, as when dealing with illumi- or with individuals, the negative nation, numbers are not absolute or positive numbers applicable, but only the Just as . electrical engineer, even if he possess a fail' knowledge of mathematics, may often find difficulty in dealing with problems, through lack of familiarity with these branches of mathematics, which have become of importance in electrical engineering, and may also find difficulty in looking up information on. curves and their evaluation, and other kindred subjects requisite for effective engineering work, Thus the following work is not intended as a complete course in mathematics, but as supplementary to the general college course of mathematics, or to the general. most efficient method of study, appears to me, to start with " Electrical Engineering Mathematics, " and after entering its third chapter, to take up the reading of the first section of " Theo- retical Elements," and then parallel the study of " Electrical ÆTHERFORCE PREFACE. vii Engineering Mathematics/ ' " Theoretical Elements of Electrical Engineering/ ' and " Theory and Calculation of Alternating Current Phenomena/' together with selected chapters from "Theory and Calculation