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{{InfoboxBook | {{InfoboxBook | ||
|title=Calculus | |title=Calculus | ||
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|isbn13=978-0471000051 | |isbn13=978-0471000051 | ||
}} | }} | ||
The textbook '''''Calculus''''' by [https://en.wikipedia.org/wiki/Tom_M._Apostol Tom Apostol] introduces calculus. It provides a rigorous treatment of theory and application, in addition to the historical context of its topics. | {{NavContainerFlex | ||
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{{NavButton|link=[[Read#Basic_Mathematics|Read]]}} | |||
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The textbook [https://simeioseismathimatikwn.files.wordpress.com/2013/03/apostol-calculusi.pdf '''''Calculus'''''] by [https://en.wikipedia.org/wiki/Tom_M._Apostol Tom Apostol] introduces calculus. It provides a rigorous treatment of theory and application, in addition to the historical context of its topics. It should be noted that there is a [https://archive.org/details/calculus-tom-m.-apostol-calculus-volume-2-2nd-edition-proper-2-1975-wiley-sons-libgen.lc/Apostol%20T.%20M.%20-%20Calculus%20vol%20II%20%281967%29/ second volume], not listed here, which covers multivariable topics and applications to subjects such as probability. | |||
== Table of Contents == | == Table of Contents == | ||
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| 1.22 || Calculation of the integral of a bounded monotonic function || 79 | | 1.22 || Calculation of the integral of a bounded monotonic function || 79 | ||
|- | |- | ||
| 1.23 || Calculation of the integral | | 1.23 || Calculation of the integral <math>\int_0^b x^p dx</math> when <math>p</math> is a positive integer || 79 | ||
|- | |- | ||
| 1.24 || The basic properties of the integral || 80 | | 1.24 || The basic properties of the integral || 80 | ||
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| 6.4 || The graph of the natural logarithm || 230 | | 6.4 || The graph of the natural logarithm || 230 | ||
|- | |- | ||
| 6.5 || Consequences of the functional equation | | 6.5 || Consequences of the functional equation <math>L(ab) = L(a) + L(b)</math> || 230 | ||
|- | |- | ||
| 6.6 || Logarithms referred to any positive base | | 6.6 || Logarithms referred to any positive base <math>b \ne 1</math> || 232 | ||
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| 6.7 || Differentiation and integration formulas involving logarithms || 233 | | 6.7 || Differentiation and integration formulas involving logarithms || 233 | ||
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| 6.13 || Exponentials expressed as powers of e || 242 | | 6.13 || Exponentials expressed as powers of e || 242 | ||
|- | |- | ||
| 6.14 || The definition of | | 6.14 || The definition of <math>e^x</math> for arbitrary real x || 244 | ||
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| 6.15 || The definition of | | 6.15 || The definition of <math>a^x</math> for <math>a > 0</math> and x real || 245 | ||
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| 6.16 || Differentiation and integration formulas involving exponentials || 245 | | 6.16 || Differentiation and integration formulas involving exponentials || 245 | ||
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| 7.13 || Exercises || 295 | | 7.13 || Exercises || 295 | ||
|- | |- | ||
| 7.14 || The symbols | | 7.14 || The symbols <math>+\inf</math> and <math>-\inf</math>. Extension of L'Hopital's rule || 296 | ||
|- | |- | ||
| 7.15 || Infinite limits || 298 | | 7.15 || Infinite limits || 298 | ||
|- | |- | ||
| 7.16 || The behavior of log | | 7.16 || The behavior of log<math>x</math> and <math>e^x</math> for large <math>x</math> || 300 | ||
|- | |- | ||
| 7.17 || Exercises || 303 | | 7.17 || Exercises || 303 | ||
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| 8.8 || Linear equations of second order with constant coefficients || 322 | | 8.8 || Linear equations of second order with constant coefficients || 322 | ||
|- | |- | ||
| 8.9 || Existence of solutions of the equation | | 8.9 || Existence of solutions of the equation <math>y^{''} + by = 0</math> || 323 | ||
|- | |- | ||
| 8.10 || Reduction of the general equation to the special case | | 8.10 || Reduction of the general equation to the special case <math>y^{''} + by = 0</math> || 324 | ||
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| 8.11 || Uniqueness theorem for the equation | | 8.11 || Uniqueness theorem for the equation <math>y^{''} + by = 0</math> || 324 | ||
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| 8.12 || Complete solution of the equation | | 8.12 || Complete solution of the equation <math>y^{''} + by = 0</math> || 326 | ||
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| 8.13 || Complete solution of the equation | | 8.13 || Complete solution of the equation <math>y^{''} + ay^{'} + by = 0</math> || 326 | ||
|- | |- | ||
| 8.14 || Exercises || 328 | | 8.14 || Exercises || 328 | ||
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| 8.15 || Nonhomogeneous linear equations of second order with constant coefficients || 329 | | 8.15 || Nonhomogeneous linear equations of second order with constant coefficients || 329 | ||
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| 8.16 || Special methods for determining a particular solution of the nonhomogeneous equation | | 8.16 || Special methods for determining a particular solution of the nonhomogeneous equation <math>y^{''} + ay^{'} + by = R</math> || 332 | ||
|- | |- | ||
| 8.17 || Exercises || 333 | | 8.17 || Exercises || 333 | ||
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| 9.3 || The complex numbers as an extension of the real numbers || 360 | | 9.3 || The complex numbers as an extension of the real numbers || 360 | ||
|- | |- | ||
| 9.4 || The imaginary unit | | 9.4 || The imaginary unit <math>i</math> || 361 | ||
|- | |- | ||
| 9.5 || Geometric interpretation. Modulus and argument || 362 | | 9.5 || Geometric interpretation. Modulus and argument || 362 | ||
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| 12.2 || The vector space of n-tuples of real numbers || 446 | | 12.2 || The vector space of n-tuples of real numbers || 446 | ||
|- | |- | ||
| 12.3 || Geometric interpretation for | | 12.3 || Geometric interpretation for <math>n \leq 3</math> || 448 | ||
|- | |- | ||
| 12.4 || Exercises || 450 | | 12.4 || Exercises || 450 | ||
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| 12.15 || Exercises || 467 | | 12.15 || Exercises || 467 | ||
|- | |- | ||
| 12.16 || The vector space | | 12.16 || The vector space <math>V_N(C)</math> of n-tuples of complex numbers || 468 | ||
|- | |- | ||
| 12.17 || Exercises || 470 | | 12.17 || Exercises || 470 | ||
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[[Category:Mathematics]] | [[Category:Mathematics]] | ||
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