Fundamental Formulas of Physics, Volume One
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The republication of this book, unabridged and corrected, fills the need for a comprehensive work on fundamental formulas of mathematical physics. It ranges from simple operations to highly sophisticated ones, all presented most lucidly with terms carefully defined and formulas given completely. In addition to basic physics, pertinent areas of chemistry, astronomy, meteorology, biology, and electronics are also included.
This is no mere listing of formulas, however. Mathematics is integrated into text, for the most part, so that each chapter stands as a brief summary or even short textbook of the field represented. The book, therefore, fills other needs than the primary function of reference and guide for research. The student will find it a handy review of familiar fields and an aid to gaining rapid insight into the techniques of new ones.
The teacher will study it as a useful guide to a broad concept of physics. The chemist, astronomer, meteorologist, biologist, and engineer will not only derive valuable aid from their special chapters, but will understand how their specialty fits into the large scheme of physics.
Vol. 1 chapter titles: Basic Mathematical Formulas, Statistics, Nomograms, Physical Constants, Classical Mechanics, Special Theory of Relativity, The General Theory of Relativity, Hydrodynamics and Aerodynamics, Boundary Value Problems in Mathematical Physics, Heat and Thermodynamics, Statistical Mechanics, Kinetic Theory of Gases: Viscosity, Thermal Conduction, and Diffusion, Electromagnetic Theory, Electronics, Sound and Acoustics.
Vol. 2 chapter titles: Geometrical Optics, Physical Optics, Electron Optics, Molecular Spectra, Atomic Spectra, Quantum Mechanics, Nuclear Theory, Cosmic Rays and High-Energy Phenomena, Particle Accelerators, Solid State, Theory of Magnetism, Physical Chemistry, Basic Formulas of Astrophysics, Celestial Mechanics, Meteorology, Biophysics.
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Fundamental Formulas of Physics, Volume One - Dover Publications
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TIME’S ARROW: THE ORIGINS OF THERMODYNAMIC BEHAVIOR, Michael C. Mackey. (0-486-43243-2)
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PREFACE
A survey of physical scientists, made several years ago, indicated the need for a comprehensive reference book on the fundamental formulas of mathematical physics. Such a book, the survey showed, should be broad, covering, in addition to basic physics, certain cross-field disciplines where physics touches upon chemistry, astronomy, meteorology, biology, and electronics.
The present volume represents an attempt to fill the indicated need. I am deeply indebted to the individual authors, who have contributed time and effort to select and assemble formulas within their special fields. Each author has had full freedom to organize his material in a form most suitable for the subject matter covered. In consequence, the styles and modes of presentation exhibit wide variety. Some authors considered a mere listing of the basic formulas as giving ample coverage. Others felt the necessity of adding appreciable explanatory text.
The independence of the authors has, inevitably, resulted in a certain amount of overlap. However, since conventional notation may vary for the different fields, the duplication of formulas should be helpful rather than confusing.
In the main, authors have emphasized the significant formulas, without attempting to develop them from basic principles. Apart from this omission, each chapter stands as a brief summary or short textbook of the field represented. In certain instances, the authors have included material not heretofore available.
The book, therefore, should fill needs other than its intended primary function of reference and guide for research. A student may find it a handy aid for review of familiar field or for gaining rapid insight into the techniques of new ones. The teacher will find it a useful guide in the broad field of physics. The chemist, the astronomer, the meteorologist, the biologist, and the engineer should derive valuable aid from the general sections as well as from the cross-field chapters in their specialties. For example, the chapter on Electromagnetic Theory has been designed to meet the needs of both engineers and physicists. The handy conversion factors facilitate rapid conversion from Gaussian to MKS units or vice versa.
In a work of this magnitude, some errors will have inevitably crept in. I should appreciate it, if readers would call them to my attention.
DONALD H. MENZEL
Harvard College Observatory
Cambridge, Mass.
Table of Contents
DOVER BOOKS ON PHYSICS
Title Page
Copyright Page
PREFACE
Chapter 1 - BASIC MATHEMATICAL FORMULAS
Chapter 2 - STATISTICS
Chapter 3 - NOMOGRAMS
Chapter 4 - PHYSICAL CONSTANTS
Chapter 5 - CLASSICAL MECHANICS
Chapter 6 - SPECIAL THEORY OF RELATIVITY
Chapter 7 - THE GENERAL THEORY OF RELATIVITY
Chapter 8 - HYDRODYNAMICS AND AERODYNAMICS
Chapter 9 - BOUNDARY VALUE PROBLEMS IN MATHEMATICAL PHYSICS
Chapter 10 - HEAT AND THERMODYNAMICS
Chapter 11 - STATISTICAL MECHANICS
Chapter 12 - KINETIC THEORY OF GASES : VISCOSITY, THERMAL CONDUCTION, AND DIFFUSION
Chapter 13 - ELECTROMAGNETIC THEORY
Chapter 14 - ELECTRONICS
Chapter 15 - SOUND AND ACOUSTICS
INDEX
Chapter 1
BASIC MATHEMATICAL FORMULAS
BY PHILIP FRANKLIN
Professor of Mathematics
Massachusetts Institute of Technology
Certain parts of pure mathematics are easily recognized as necessary tools of theoretical physics, and the results most frequently applied may be summarized as a compilation of formulas and theorems. But any physicist who hopes or expects that any moderate sized compilation of mathematical results will satisfy all his needs is doomed to disappointment. In fact no collection, short of a library, could begin to fulfill all the demands that the physicist will make upon the pure mathematician. And these demands grow constantly with the increasing size of the field of mathematical physics.
The following compilation of formulas is, therefore, intended to be representative rather than comprehensive. To conserve space, certain elementary formulas and extensive tables of indefinite integrals have been omitted, since these are available in many well-known mathematical handbooks.
A select list of reference books, arranged by subject matter, is given at the end of Chapter 1. This bibliography may assist the reader in checking formulas, or in pursuing details or extensions beyond the material given here.
As a preliminary check on the formulas to be included, a tentative list was submitted to a number of physicists. These included all the authors of the various chapters of this book. They contributed numerous suggestions, which have been followed as far as space requirements have permitted.
1. Algebra
1.1. Quadratic equations. If a ≠ 0, the roots of
(1)
are
(2)
(3)
When b|4ac|, the second form for x1 when b > 0, and for x2 when b < 0 is easier to compute with precision, since it involves the sum, instead of the difference, of two nearly equal terms.
1.2. Logarithms. Let In represent natural logarithm, or loge where e = 2.71828. Then for M > 0, N > 0 :
(1.)
In this book log10 will be written simply log. For conversion between base e and base 10
(2)
1.3. Binomial theorem. For n any positive integer,
(1)
or
(2)
where n! (factorial n) is defined by
(3)
1.4. Multinomial theorem. For n any positive integer, the general term in the expansion of (a1 + a2 + ... +ak)n is
(1)
where r1, r2, ... rk are positive integers such that r1 + r2 + ... + rk = n.
1.5. Proportion
(1)
If a/A = b/B = c/C = k, then for any weighting factors p, q, r each fraction equals
(2)
1.6. Progressions. Let l be the last term. Then the sum of the arithmetic progression to n terms
s = a + (a + d) + (a + 2d) + ... + [a + (n - 1)d]
is
(1)
where l = a + (n — 1)d, and the sum of the geometric progression to n terms
(2)
1.7. Algebraic equations. The general equation of the nth degree
(1)
has n roots. If the roots of P(x) = 0, or zeros of P(x), are r1, r2, ..., rn, then
P(x) = a0(x — r1) (x — r2) ... (x — rn)
and the symmetric functions of the roots
(2)
If m roots are equal to r, then r is a multiple root of order m. In this case P(x) = (x r)mQ1(x) and P′(x) = (x — r)m ¹Q2(x), where P’(x) = dP/dx, § 3.1. Thus r will be a multiple root of P’(x) = 0 of order m — 1. All multiple or repeated roots will be zeros of the greatest common divisor of P(x) and P’(x).
If r is known to be a zero of P(x), so that P(r) = 0, then (x — r) is a factor of P(r), and dividing P(x) by (x — r) will lead to a depressed equation with degree lowered by one.
When P(a) and P(b) have opposite signs, a real root lies between a and b. The interpolated value c = a P(a) (b — a)/[P(b) — P(a)], and calculation of P(c) will lead to new closer limits.
If c1 is an approximate root, c2 = cP(c)/P′(c) is Newton’s improved approximation. We may repeat this process. Newton’s procedure can be applied to transcendental equations. If an approximate complex root can be found, it can be improved by Newton’s method.
1.8. Determinants. The determinant of the nth order
(1)
is defined to be the sum of n! terms
(2)
In each term the second subscripts ijk...l are one order or permutation of the numbers 123...n. The even permutations, which contain an even number of inversions, are given the plus sign. The odd permutations, which contain an odd number of inversions, are given the minus sign.
The cofactor Aij of the element aij 1)i+j times the determinant of the (n 1)st order obtained from D by deleting the ith row and jth column.
The value of a determinant D is unchanged if the corresponding rows and columns are interchanged, or if, to each element of any row (or column), is added m times the corresponding element in another row (or column).
If any two rows (or columns) are interchanged, D 1). If each element of any one row (or column) is multiplied by m, then D is multiplied by m.
If any two rows (or columns) are equal, or proportional, D = 0.
(3)
(4)
where j and k are any two of the integers 1, 2, ..., n.
1.9. Linear equations. The solution of the system
(1)
is unique if the determinant of § 1.8, D ≠ 0. The solution is
(2)
where Ck is the nth order determinant obtained from D by replacing the elements of its kth column a1k, a2k, ..., ank by c1, c2, ..., cn.
When all the cj = 0, the system is homogeneous. In this case, if D = 0, but not all the Aij are zero, the ratios of the xi satisfy
(3)
A homogeneous system has no nonzero solutions if D ≠ 0.
2. Trigonometry
2.1. Angles. Angles are measured either in degrees or in radians, units such that
2 right angles = 180 degrees =π radians
where π = 3.14159. For conversion between degrees and radians,
1 degree = 0.017453 radian; 1 radian = 57.296 degrees
2.2. Trigonometric functions. For a single angle A, the equations
(1)
define the tangent, cotangent, secant, and cosecant in terms of the sine and cosine. Between the six functions we have the relations
(2)
2.3. Functions of sums and differences
(1)
(2)
2.4. Addition theorems
(1)
2.5. Multiple angles
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
2.6. Direction cosines. The direction cosines of a line in space are the cosines of the angles α, β, y which a parallel line through the origin O makes with the coordinate axes OX, OY, OZ. For the line segment P1P2, joining the points P1 = (x1,y1,z1) and P2 = (x2,y2,z2) the direction cosines are
(1)
where d is the distance between P1 and P2, so that
(2)
For any set of direction cosines, cos² α + cos² β + cos² γ = 1.
If a, b, c are direction ratios for a line, or numbers proportional to the direction cosines, then
(3)
The angle θ between two lines with direction angles α1, β1, and α2, β2, γ2 may be found from
(4)
The equation of a plane is Ax + By + Cz = D. A, B, C are direction ratios for any normal line perpendicular to the plane. The distance from the plane to P1 = (x1,y1,z1) is
(5)
The equations of a straight line with direction. ratios A, B, C through the point P1 = (x1,y1,z1) are
(6)
The angle θ between two planes whose normals have direction ratios] A1, B1, C1, and A2, B2, C2, or between two lines with these direction ratios may be found from
(7)
2.7. Plane right triangle. C = 90°, opposite c.
(1)
2.8. Amplitude and phase
(1)
where the amplitude c and phase A are related to the constants a and b by the equations of §2.7.
2.9. Plane oblique triangle. Sides a, b, c opposite A, B, C.
A + B + C = 180 degrees or π radians
Law of sines :
(1)
Law of cosines :
(2)
(3)
where s (a + b + c).
2.10. Spherical right triangle. C = 90°, opposite c.
(1)
(2)
(3)
(4)
(5)
2.11. Spherical oblique triangle. Sides a, b, c opposites A, B, C.
0° < a + b + c < 360°, 180° < A + B + C < 540°
Law of sines :
(1)
Law of cosines :
(2)
(3)
Spherical excess :
E = A + B + C 180°
If R is the radius of the sphere upon which the triangle lies, its area K = πR²E/180. If
s (a + b + c),
(4)
2.12. Hyperbolic functions. For a single number x, the equations
(1)
define the hyperbolic sine and hyperbolic cosine in terms of the exponential function. And the equations
(2)
define the hyperbolic tangent, hyperbolic cotangent, hyperbolic secant and hyperbolic cosecant, respectively, in terms of the hyperbolic sine and hyperbolic cosine. The four functions sinh x, tanh x, coth x, and csch x x) = —sinh x xtanh x. The functions cosh x and sech x x) = cosh x. Between the six hyperbolic functions we have the relations
(3)
2.13. Functions of sums and differences
(1)
2.14. Multiple arguments
(1)
(2)
2.15. Sine, cosine, and complex exponential function. i² = 1.
(1)
(2)
2.16. Trigonometric and hyperbolic. functions
(1)
2.17. Sine and cosine of complex arguments
(1)
(2)
(3)
2.18. Inverse functions and logarithms
(1)
(2)
(3)
3. Differential Calculus
3.1. The derivative. Let y = f(x) be a function of x. Then the derivative dy/dx is defined by the equation
(1)
Alternative notations for the derivative are f′(x), Dxy, yis used for dy/dt, when y = F(t).
3.2. Higher derivatives. The second derivative is
(1)
The nth derivative is
(2)
3.3. Partial derivatives. Let z = f(x,y) be a function of two variables. Then the partial derivative ∂z/∂x is defined by the equation
(1)
Similarly the derivative ∂z/∂y is formed with y varying and x fixed.
The second partial derivatives are defined by
(2)
The same process defines partial derivatives of any order, and when the highest derivatives involved are continuous, the result is independent of the order in which the differentiations are performed.
For a function of more than two variables, we may form partial derivatives by varying the independent quantities one at a time. Thus for u = f(x,y,z) the first derivatives are ∂u/∂x, ∂u/∂y, ∂u/∂z.
3.4. Derivatives of functions
Inverse functions :
(1)
Chain rule:
(2)
Implicit function:
(3)
Parameter:
y = y(t), x = x(t)
Let
(4)
Then
Linearity:
(5)
3.5. Products
(1)
Leibniz rule :
(2)
The nCr are defined in §1.3.
3.6. Powers and quotient
(1)
3.7. Logarithmic differentiation
(1)
(2)
(3)
3.8. Polynomials
(1)
(2)
3.9. Exponentials and logarithms. We write In for log,, where e = 2.71828, as in §1.2.
(1)
(2)
3.10. Trigonometric functions
(1)
(2)
3.11. Inverse trigonometric functions
(1)
(2)
3.12. Hyperbolic functions
(1)
(2)
3.13. Inverse hyperbolic functions
(1)
(2)
3.14. Differential. Let y = f(x), and Δx be the increment in x. Then
(1)
Parameter:
(2)
First differentials are independent of the choice of independent variable.
3.15. Total differential. For two independent variables z = f(x, y),
(1)
For three variables, u = f(x,y,z),
(2)
(3)
and similarly for functions of more variables.
3.16. Exact differential. The condition that as in § 3.15 for some f(x,y), with z = f(x,y) is
A(x,y)dx + B(x,y)dy = dz,
(1)
for some f(x,y,z) to make
A(x,y,z)dx + B(x,y,z)dy + C(x,y,z)dz = du,
with u = f(x,y,z), the condition is
(2)
3.17. Maximum and minimum values. Let y = f(x) be regular in the interval a, b. Then at a relative maximum, f(x1) with a < x1 < b, f’(x) decreases from plus to minus as x increases through x1. At a relative minimum, f(x2) with a < x2 < b, f′(x) increases from minus to plus as x increases through x2. Thus the largest and smallest values of f(x) will be included in the set f(a), f(b), and f(xk), where f′(xk) = 0.
3.18. Points of inflection. The graph of y = f(x) is concave downward in any interval throughout which f′′(x) is negative. The graph is concave upward in any interval throughout which f′′(x) is positive. At any point y3 = f(x3) such that f′′(x) changes sign as x increases through x3, the graph of y = f(x) is said to have a point of inflection.
3.19. Increasing absolute value. Let OP = s(t), the distance along OX. Then the velocity v = ds/dt , and the acceleration a = dv/dt = d²s/dt² = s. Then s increases when v is positive, and v increases when a is positive. The distance from O, or |s| increases when |s|² = s² increases, so that |s| increases when s is positive, or when s and v have the same sign. The speed or |v| increases when |v|² = v² increases, so that |v| is positive, or when v and a have the same sign.
3.20. Arc length. In the triangle formed by dx, dy, and ds, the differential of arc length, the angle opposite dy is the slope angle, and the angle opposite ds is 90°. Thus
(1)
(2)
In polar coordinates, r = r(θ),
ds² = dr² + r²dθ²
and
(3)
3.21. Curvature. R = ds/dτ is the radius of curvature, and the curvature K = 1/R is given by
(1)
In polar coordinates r = r(θ),
(2)
3.22. Acceleration in plane motion. The velocity vector has x and y = dx/dt = dy/dt, . The acceleration vector has x and y components x = d²x/dt² and ÿ = d²y/dt². = dv/dt and a normal component v²/R. As in §3.21, 1/R = K. The magnitude of the acceleration is
(1)
3.23. Theorem of the mean. Let f(x) and f(x) be regular in the interval a, b. Then for at least one value xk with a < xk < b,
Rolle’s theorem:
If
(1)
Law of the mean:
(2)
Cauchy’s mean value theorem:
If
(3)
3.24. Indeterminate forms. Let f(x) and f(x) each approach zero as x approaches a. We briefly describe the evaluation of lim f(x)/F(x) as the indeterminate form 0/0. When f(x) and f(xa) can be easily found, the limit may be found by using these series.
l’Hospital’s rule: If the limit on the right exists, then when f(x) → 0, F(x) → 0,
(1)
In this rule x may approach a from one side, x→a+ or x→ a , and it applies when in place of x→a, x→+∞, or x∞. The rule in any form also applies when as x → a, f(x) and F(x) each tend to infinity. This is the indeterminate from ∞/∞.
By writing f = 1/(1/f∞ to 0/0.
If the evaluation of lim fF leads to an indeterminate form 0⁰, 1∞ ∞⁰, the evaluation of L = lim F ln f 0. This may be found as indicated above, and then lim fF = eL.
3.25. Taylor’s theorem. Let f(x) be analytic at a. Then
(1)
For real values, the remainder after the term with f(n ¹)(a) is
(2)
where x1 is a suitably chosen value in the interval a, x. An alternative form is
(3)
The special case when a = 0 is called Maclaurin’s series.
(4)
For computation these series are usually used with (x a) or h small, and the remainder has the order of magnitude of the first term neglected.
Let f(x,y) be analytic at (a,b). Then in the notation of §3.3,
(5)
An alternative form is
(6)
The special case when a = 0, b = 0, Maclaurin’s series is
(7)
And similar expansions hold for any number of variables.
3.26. Differentiation of integrals
(1)
(2)
(3)
If f(x,x) is infinite or otherwise singular one may use
(4)
(5)
4. Integral Calculus
4.1. Indefinite integral. With respect to x, the indefinite integral of f(x) is f(x) provided that