| Introduction |
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1 | (2) |
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Formulas, Graphs and Tables |
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3 | (218) |
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Basic formulas of elementary mathematics |
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3 | (42) |
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3 | (2) |
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5 | (2) |
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Area and circumference of plane figures |
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7 | (3) |
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Volume and surface area of solids |
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10 | (3) |
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Volumes and surface areas of regular polyhedra |
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13 | (2) |
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Volume and surface area of n-dimensional balls |
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15 | (1) |
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Basic formulas for analytic geometry in the plane |
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16 | (9) |
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Basic formulas of analytic geometry of space |
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25 | (1) |
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Powers, roots and logarithms |
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26 | (2) |
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Elementary algebraic formulas |
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28 | (8) |
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36 | (5) |
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Application to the motion of the planets |
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41 | (4) |
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Elementary functions and graphs |
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45 | (29) |
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Transformation of functions |
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47 | (1) |
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48 | (1) |
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49 | (1) |
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50 | (1) |
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50 | (2) |
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52 | (1) |
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The general exponential function |
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53 | (1) |
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53 | (6) |
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59 | (4) |
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The hyperbolic functions sinh x and cosh x |
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63 | (1) |
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The hyperbolic functions tanh x and coth x |
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64 | (2) |
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The inverse trigonometric functions |
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66 | (2) |
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The inverse hyperbolic functions |
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68 | (2) |
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70 | (1) |
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71 | (3) |
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Mathematics and computers -- a revolution in mathematics |
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74 | (1) |
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Tables of mathematical statistics |
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75 | (23) |
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Empirical data for sequences of measurements (trials) |
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75 | (2) |
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The theoretical distribution function |
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77 | (2) |
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Checking for a normal distribution |
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79 | (1) |
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The statistical evaluation of a sequence of measurements |
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80 | (1) |
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The statistical comparison of two sequences of measurements |
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80 | (3) |
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Tables of mathematical statistics |
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83 | (15) |
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Tables of values of special functions |
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98 | (12) |
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The gamma functions Γ(x) and 1/γ(x) |
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98 | (1) |
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Cylinder functions (also known as Bessel functions) |
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99 | (4) |
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Spherical functions (Legendre polynomials) |
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103 | (1) |
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104 | (2) |
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Integral trigonometric and exponential functions |
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106 | (2) |
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108 | (1) |
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108 | (1) |
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Changing from degrees to radians |
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109 | (1) |
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Table of prime numbers ≤ 4000 |
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110 | (1) |
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Formulas for series and products |
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111 | (22) |
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111 | (3) |
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114 | (10) |
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124 | (3) |
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127 | (5) |
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132 | (1) |
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Tables for differentiation of functions |
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133 | (5) |
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Differentiation of elementary functions |
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133 | (2) |
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Rules for differentiation of functions of one variable |
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135 | (1) |
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Rules for differentiating functions of several variables |
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136 | (2) |
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138 | (54) |
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Integration of elementary functions |
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138 | (2) |
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140 | (4) |
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Integration of rational functions |
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144 | (1) |
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145 | (4) |
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Tables of indefinite integrals |
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149 | (37) |
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Tables of definite integrals |
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186 | (6) |
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Tables on integral transformations |
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192 | (29) |
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192 | (13) |
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205 | (16) |
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221 | (378) |
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222 | (16) |
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222 | (6) |
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228 | (5) |
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Applications to oscillations |
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233 | (1) |
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Calculations with equalities |
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234 | (2) |
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Calculations with inequalities |
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236 | (2) |
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238 | (11) |
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238 | (1) |
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The Hilbert axioms for the real numbers |
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239 | (3) |
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Sequences of real numbers |
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242 | (3) |
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Criteria for convergence of sequences |
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245 | (4) |
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249 | (13) |
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Functions of a real variable |
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249 | (5) |
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Metric spaces and point sets |
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254 | (5) |
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Functions of several variables |
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259 | (3) |
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Differentiation of functions of a real variable |
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262 | (16) |
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262 | (2) |
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264 | (1) |
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Increasing and decreasing functions |
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265 | (1) |
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266 | (2) |
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Taylor's theorem and the local behavior of functions |
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268 | (9) |
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277 | (1) |
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Derivatives of functions of several real variables |
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278 | (28) |
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278 | (1) |
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279 | (3) |
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282 | (3) |
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Applications to the transformation of differential operators |
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285 | (2) |
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Application to the dependency of functions |
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287 | (1) |
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The theorem on implicit functions |
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288 | (2) |
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290 | (2) |
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The nth variation and Taylor's theorem |
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292 | (1) |
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Applications to estimation of errors |
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293 | (2) |
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295 | (11) |
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Integration of functions of a real variable |
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306 | (15) |
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307 | (3) |
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Existence of the integral |
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310 | (2) |
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The fundamental theorem of calculus |
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312 | (1) |
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313 | (1) |
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314 | (3) |
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Integration on unbounded intervals |
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317 | (1) |
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Integration of unbounded functions |
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318 | (1) |
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The Cauchy principal value |
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318 | (1) |
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Application to arc length |
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319 | (1) |
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A standard argument from physics |
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320 | (1) |
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Integration of functions of several real variables |
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321 | (30) |
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321 | (8) |
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Existence of the integral |
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329 | (3) |
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Calculations with integrals |
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332 | (1) |
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The principle of Cavalieri (iterated integration) |
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333 | (2) |
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335 | (1) |
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The fundamental theorem of calculus (theorem of Gauss-Stokes) |
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335 | (6) |
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The Riemannian surface measure |
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341 | (2) |
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343 | (1) |
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344 | (4) |
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Applications to the center of mass and center of inertia |
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348 | (2) |
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Integrals depending on parameters |
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350 | (1) |
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351 | (6) |
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Linear combinations of vectors |
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351 | (2) |
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353 | (1) |
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Multiplication of vectors |
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354 | (3) |
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Vector analysis and physical fields |
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357 | (19) |
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Velocity and acceleration |
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357 | (2) |
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Gradient, divergence and curl |
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359 | (2) |
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Applications to deformations |
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361 | (2) |
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Calculus with the nabla operator |
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363 | (3) |
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Work, potential energy and integral curves |
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366 | (2) |
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Applications to conservation laws in mechanics |
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368 | (2) |
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Flows, conservation laws and the integral theorem of Gauss |
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370 | (2) |
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The integral theorem of Stokes |
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372 | (1) |
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Main theorem of vector analysis |
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373 | (1) |
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Application to Maxwell's equations in electromagnetism |
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374 | (2) |
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Cartan's differential calculus |
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376 | (1) |
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376 | (15) |
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378 | (2) |
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Calculations with infinite series |
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380 | (2) |
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382 | (3) |
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385 | (4) |
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Summation of divergent series: |
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389 | (1) |
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389 | (2) |
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391 | (16) |
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The Laplace transformation |
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393 | (5) |
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The Fourier transformation |
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398 | (5) |
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403 | (4) |
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Ordinary differential equations |
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407 | (61) |
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407 | (8) |
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415 | (9) |
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The classification of differential equations |
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424 | (10) |
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Elementary methods of solution |
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434 | (16) |
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450 | (4) |
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Systems of linear differential equations and the propagator |
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454 | (3) |
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457 | (2) |
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Boundary value problems and Green's functions |
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459 | (5) |
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464 | (4) |
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Partial differential equations |
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468 | (64) |
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Equations of first order of mathematical physics |
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469 | (27) |
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Equations of mathematical physics of the second order |
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496 | (15) |
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The role of characteristics |
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511 | (10) |
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General principles for uniqueness |
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521 | (1) |
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General existence results |
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522 | (10) |
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532 | (67) |
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533 | (1) |
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Sequences of complex numbers |
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534 | (1) |
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535 | (2) |
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537 | (4) |
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The language of differential forms |
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541 | (2) |
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Representations of functions |
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543 | (6) |
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The calculus of residues and the calculation of integrals |
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549 | (2) |
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551 | (1) |
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Applications to the fundamental theorem of algebra |
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552 | (2) |
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Biholomorphic maps and the Riemann mapping theorem |
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554 | (1) |
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Examples of conformal maps |
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555 | (8) |
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Applications to harmonic functions |
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563 | (3) |
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Applications to hydrodynamics |
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566 | (2) |
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Applications in electrostatics and magnetostatics |
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568 | (1) |
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Analytic continuation and the identity principle |
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569 | (3) |
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Applications to the Euler gamma function |
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572 | (2) |
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Elliptic functions and elliptic integrals |
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574 | (7) |
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Modular forms and the inversion problem for the function |
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581 | (3) |
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584 | (8) |
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Singular differential equations |
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592 | (1) |
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The Gaussian hypergeometric differential equation |
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593 | (1) |
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Application to the Bessel differential equation |
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593 | (2) |
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Functions of several complex variables |
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595 | (4) |
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599 | (126) |
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599 | (27) |
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599 | (3) |
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602 | (3) |
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605 | (5) |
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Systems of linear equations |
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610 | (5) |
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Calculations with polynomials |
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615 | (3) |
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The fundamental theorem of algebra according to Gauss |
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618 | (6) |
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Partial fraction decomposition |
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624 | (2) |
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626 | (11) |
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626 | (2) |
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Normal forms for matrices |
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628 | (7) |
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635 | (2) |
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637 | (13) |
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637 | (1) |
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638 | (3) |
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641 | (4) |
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Calculating with linear spaces |
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645 | (3) |
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648 | (2) |
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650 | (13) |
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650 | (1) |
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Calculations with multilinear forms |
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651 | (6) |
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657 | (4) |
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661 | (1) |
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662 | (1) |
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663 | (12) |
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663 | (6) |
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669 | (3) |
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672 | (3) |
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Galois theory and algebraic equations |
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675 | (10) |
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The three famous ancient problems |
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675 | (1) |
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The main theorem of Galois theory |
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675 | (3) |
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The generalized fundamental theorem of algebra |
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678 | (1) |
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Classification of field extensions |
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679 | (1) |
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The main theorem on equations which can be solved by radicals |
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680 | (2) |
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Constructions with a ruler and a compass |
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682 | (3) |
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685 | (40) |
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686 | (1) |
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687 | (3) |
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The distribution of prime numbers |
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690 | (6) |
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696 | (3) |
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The approximation of irrational numbers by rational numbers and continued fractions |
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699 | (6) |
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705 | (3) |
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Applications to the number π |
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708 | (4) |
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712 | (3) |
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Minkowski's geometry of numbers |
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715 | (1) |
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The fundamental local--global principle in number theory |
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715 | (2) |
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Ideals and the theory of divisors |
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717 | (2) |
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Applications to quadratic number fields |
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719 | (2) |
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The analytic class number formula |
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721 | (1) |
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Hilbert's class field theory for general number fields |
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722 | (3) |
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725 | (148) |
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The basic idea of geometry epitomized by Klein's Erlanger Program |
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725 | (1) |
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726 | (23) |
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726 | (7) |
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733 | (3) |
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736 | (5) |
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Applications to sea and air travel |
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741 | (1) |
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The Hilbert axioms of geometry |
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742 | (3) |
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The parallel axiom of Euclid |
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745 | (1) |
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The non-Euclidean elliptic geometry |
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746 | (1) |
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The non-Euclidean hyperbolic geometry |
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747 | (2) |
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Applications of vector algebra in analytic geometry |
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749 | (4) |
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750 | (1) |
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Lines and planes in space |
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751 | (1) |
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752 | (1) |
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Euclidean geometry (geometry of motion) |
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753 | (7) |
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The group of Euclidean motions |
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753 | (1) |
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754 | (1) |
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755 | (5) |
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760 | (9) |
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760 | (2) |
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762 | (1) |
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The n-dimensional real projective space |
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763 | (2) |
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The n-dimensional complex projective space |
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765 | (1) |
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The classification of plane geometries |
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765 | (4) |
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769 | (19) |
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770 | (5) |
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775 | (3) |
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The Gaussian local theory of surfaces |
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778 | (10) |
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Gauss' global theory of surfaces |
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788 | (1) |
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788 | (11) |
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788 | (1) |
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789 | (1) |
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790 | (1) |
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Huygens' tractrix and the catenary curve |
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790 | (1) |
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The lemniscate of Jakob Bernoulli and Cassini's oval |
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791 | (2) |
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793 | (1) |
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793 | (1) |
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794 | (2) |
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796 | (3) |
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799 | (38) |
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799 | (9) |
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808 | (5) |
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Applications to the calculation of integrals |
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813 | (1) |
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The projective complex form of a plane algebraic curve |
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814 | (4) |
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818 | (4) |
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822 | (6) |
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Analytic sets and the Weierstrass preparation theorem |
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828 | (1) |
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The resolution of singularities |
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829 | (2) |
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The algebraization of modern algebraic geometry |
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831 | (6) |
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Geometries of modern physics |
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837 | (36) |
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837 | (3) |
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Unitary geometry. Hilbert spaces and elementary particles |
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840 | (7) |
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847 | (3) |
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850 | (4) |
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Applications to the special theory of relativity |
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854 | (6) |
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Spin geometry and fermions |
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860 | (8) |
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Almost complex structures |
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868 | (1) |
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869 | (4) |
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Foundations of Mathematics |
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873 | (36) |
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The language of mathematics |
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873 | (5) |
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True and false statements |
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873 | (1) |
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874 | (2) |
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Tautological and logical laws |
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876 | (2) |
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878 | (6) |
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878 | (1) |
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878 | (1) |
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879 | (1) |
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879 | (2) |
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The necessity of proofs in the age of computers |
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881 | (1) |
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882 | (2) |
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884 | (11) |
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884 | (2) |
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886 | (3) |
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889 | (2) |
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891 | (1) |
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892 | (3) |
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895 | (1) |
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895 | (10) |
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896 | (3) |
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899 | (1) |
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900 | (1) |
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Cantor's structure at infinity |
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901 | (4) |
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The history of the axiomatic method |
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905 | (4) |
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Calculus of Variations and Optimization |
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909 | (66) |
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Calculus of variations one variable |
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910 | (25) |
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The Euler-Lagrange equations |
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910 | (3) |
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913 | (6) |
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919 | (6) |
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925 | (2) |
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Sufficient conditions for a local minimum |
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927 | (3) |
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Problems with constraints and Lagrange multipliers |
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930 | (1) |
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931 | (3) |
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Natural boundary conditions |
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934 | (1) |
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Calculus of variations -- several variables |
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935 | (5) |
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The Euler-Lagrange equations |
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935 | (1) |
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936 | (3) |
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Problems with constraints and Lagrange multipliers |
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939 | (1) |
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940 | (6) |
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Bellman dynamical optimization |
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941 | (1) |
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942 | (1) |
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The Pontryagin maximum principle |
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943 | (1) |
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944 | (2) |
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Classical non-linear optimization |
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946 | (6) |
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Local minimization problems |
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946 | (1) |
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Global minimization problems and convexity |
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947 | (1) |
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Applications to Gauss' method of least squares |
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947 | (1) |
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Applications to pseudo-inverses |
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948 | (1) |
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Problems with constraints and Lagrange multipliers |
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948 | (2) |
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950 | (1) |
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951 | (1) |
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Duality theory and saddle points |
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951 | (1) |
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952 | (11) |
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952 | (3) |
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The general linear optimization problem |
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955 | (2) |
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The normal form of an optimization problem and the minimal test |
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957 | (1) |
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958 | (1) |
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958 | (3) |
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Obtaining the normal form |
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961 | (1) |
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Duality in linear optimization |
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962 | (1) |
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Modifications of the simplex algorithm |
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963 | (1) |
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Applications of linear optimization |
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963 | (12) |
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963 | (1) |
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964 | (1) |
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Distributing resources or products |
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964 | (1) |
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965 | (1) |
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Linear transportation problems |
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966 | (9) |
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Stochastic Calculus -- Mathematics of Chance |
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975 | (74) |
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976 | (13) |
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The classical probability model |
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977 | (2) |
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The law of large numbers due to Jakob Bernoulli |
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|
979 | (1) |
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The limit theorem of de Moivre |
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980 | (1) |
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The Gaussian normal distribution |
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980 | (3) |
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The correlation coefficient |
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|
983 | (3) |
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Applications to classical statistical physics |
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|
986 | (3) |
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Kolmogorov's axiomatic foundation of probability theory |
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989 | (26) |
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Calculations with events and probabilities |
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992 | (3) |
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|
995 | (6) |
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1001 | (4) |
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1005 | (2) |
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The Bernoulli model for successive independent trials |
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1007 | (8) |
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1015 | (16) |
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1016 | (1) |
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1017 | (1) |
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Investigating normally distributed measurements |
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1018 | (3) |
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The empirical distribution function |
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|
1021 | (6) |
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The maximal likelihood method |
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1027 | (2) |
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1029 | (2) |
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1031 | (18) |
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1033 | (6) |
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Markov chains and stochastic matrices |
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1039 | (2) |
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1041 | (1) |
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Brownian motion and diffusion |
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1042 | (4) |
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The main theorem of Kolmogorov for general stochastic processes |
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|
1046 | (3) |
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Numerical Mathematics and Scientific Computing |
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1049 | (130) |
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Numerical computation and error analysis |
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1050 | (5) |
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1050 | (1) |
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Representing numbers on computers |
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1051 | (1) |
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Sources of error, finding errors, condition and stability |
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1052 | (3) |
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1055 | (20) |
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Linear systems of equations - direct methods |
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1055 | (7) |
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Iterative solutions of linear systems of equations |
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1062 | (3) |
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|
1065 | (4) |
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Fitting and the method of least squares |
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1069 | (6) |
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1075 | (18) |
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Interpolation polynomials |
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|
1075 | (9) |
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Numerical differentiation |
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|
1084 | (1) |
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1085 | (8) |
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|
1093 | (9) |
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|
1093 | (1) |
|
Non-linear systems of equations |
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|
1094 | (3) |
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Determination of zeros of polynomials |
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1097 | (5) |
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|
1102 | (7) |
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Approximation in quadratic means |
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|
1102 | (4) |
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|
1106 | (2) |
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Approximate uniform approximation |
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|
1108 | (1) |
|
Ordinary differential equations |
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|
1109 | (12) |
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1109 | (9) |
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|
1118 | (3) |
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Partial differential equations |
|
|
1121 | (58) |
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|
1121 | (1) |
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An overview of discretization procedures |
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|
1122 | (5) |
|
Elliptic differential equations |
|
|
1127 | (11) |
|
Parabolic differential equations |
|
|
1138 | (3) |
|
Hyperbolic differential equations |
|
|
1141 | (8) |
|
Adaptive discretization procedures |
|
|
1149 | (3) |
|
Iterative solutions of systems of equations |
|
|
1152 | (11) |
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|
|
1163 | (2) |
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|
1165 | (11) |
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|
|
1176 | (3) |
| Sketch of the history of mathematics |
|
1179 | (24) |
| Bibliography |
|
1203 | (28) |
| List of Names |
|
1231 | (4) |
| Index |
|
1235 | (40) |
| Mathematical symbols |
|
1275 | (4) |
| Dimensions of physical quantities |
|
1279 | (2) |
| Tables of physical constants |
|
1281 | |
Other versions by this Author
