Editing Calculus (Book)
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{{Stub}} | |||
{{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. | |||
Β | |||
The textbook | |||
== Table of Contents == | == Table of Contents == | ||
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| 1.2 || Functions. Informal description and examples || 50 | | 1.2 || Functions. Informal description and examples || 50 | ||
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| | | 1.3 || Functions. Formal definition as a set of ordered pairs || 53 | ||
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| 1.4 || More examples of real functions || 54 | | 1.4 || More examples of real functions || 54 | ||
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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 | ||
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| 1.23 || Calculation of the integral | | 1.23 || Calculation of the integral \(\int_0^b x^p dx\) when \(p\) is a positive integer || 79 | ||
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| 1.24 || The basic properties of the integral || 80 | | 1.24 || The basic properties of the integral || 80 | ||
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| 4.21 || Exercises || 194 | | 4.21 || Exercises || 194 | ||
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| | | 4.22 || Partial derivatives || 196 | ||
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| | | 4.23 || Exercises || 201 | ||
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! colspan="3" | 5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION | ! colspan="3" | 5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION | ||
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| 5.10 || Exercises || 220 | | 5.10 || Exercises || 220 | ||
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| | | 5.11 || Miscellaneous review exercises || 222 | ||
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! colspan="3" | 6. THE LOGARITHM, THE EXPONENTIAL, AND THE INVERSE TRIGONOMETRIC FUNCTIONS | ! colspan="3" | 6. THE LOGARITHM, THE EXPONENTIAL, AND THE INVERSE TRIGONOMETRIC FUNCTIONS | ||
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| 6.4 || The graph of the natural logarithm || 230 | | 6.4 || The graph of the natural logarithm || 230 | ||
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| 6.5 || Consequences of the functional equation | | 6.5 || Consequences of the functional equation \(L(ab) = L(a) + L(b)\) || 230 | ||
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| 6.6 || Logarithms referred to any positive base | | 6.6 || Logarithms referred to any positive base \(b \ne 1\) || 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 | ||
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| 6.14 || The definition of | | 6.14 || The definition of \(e^x\) for arbitrary real x || 244 | ||
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| 6.15 || The definition of | | 6.15 || The definition of \(a^x\) for \(a > 0\) 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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| 7.6 || Estimates for the error in Taylor's formula || 280 | | 7.6 || Estimates for the error in Taylor's formula || 280 | ||
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| | | 7.7 || Other forms of the remainder in Taylor's formula || 283 | ||
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| 7.8 || Exercises || 284 | | 7.8 || Exercises || 284 | ||
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| 7.13 || Exercises || 295 | | 7.13 || Exercises || 295 | ||
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| 7.14 || The symbols | | 7.14 || The symbols \(+\inf\) and \(-\inf\). Extension of L'Hopital's rule || 296 | ||
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| 7.15 || Infinite limits || 298 | | 7.15 || Infinite limits || 298 | ||
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| 7.16 || The behavior of log | | 7.16 || The behavior of log\(x\) and \(e^x\) for large \(x\) || 300 | ||
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| 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 | ||
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| 8.9 || Existence of solutions of the equation | | 8.9 || Existence of solutions of the equation \(y^{''} + by = 0\) || 323 | ||
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| 8.10 || Reduction of the general equation to the special case | | 8.10 || Reduction of the general equation to the special case \(y^{''} + by = 0\) || 324 | ||
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| 8.11 || Uniqueness theorem for the equation | | 8.11 || Uniqueness theorem for the equation \(y^{''} + by = 0\) || 324 | ||
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| 8.12 || Complete solution of the equation | | 8.12 || Complete solution of the equation \(y^{''} + by = 0\) || 326 | ||
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| 8.13 || Complete solution of the equation | | 8.13 || Complete solution of the equation \(y^{''} + ay^' + by = 0\) || 326 | ||
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| 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 \(y^{''} + ay^' + by = R\) || 332 | ||
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| 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 | ||
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| 9.4 || The imaginary unit | | 9.4 || The imaginary unit \(i\) || 361 | ||
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| 9.5 || Geometric interpretation. Modulus and argument || 362 | | 9.5 || Geometric interpretation. Modulus and argument || 362 | ||
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| 10.9 || Exercises || 391 | | 10.9 || Exercises || 391 | ||
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| | | 10.10 || Exercises on decimal expansions || 393 | ||
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| 10.11 || Tests for convergence || 394 | | 10.11 || Tests for convergence || 394 | ||
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| 10.20 || Exercises || 409 | | 10.20 || Exercises || 409 | ||
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| | | 10.21 || Rearrangements of series || 411 | ||
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| 10.22 || Miscellaneous review exercises || 414 | | 10.22 || Miscellaneous review exercises || 414 | ||
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| 11.11 || Power-series expansions for the exponential and trigonometric functions || 435 | | 11.11 || Power-series expansions for the exponential and trigonometric functions || 435 | ||
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| | | 11.12 || Bernstein's theorem || 437 | ||
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| 11.13 || Exercises || 438 | | 11.13 || Exercises || 438 | ||
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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 | ||
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| 12.3 || Geometric interpretation for | | 12.3 || Geometric interpretation for \(n \leq 3\) || 448 | ||
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| 12.4 || Exercises || 450 | | 12.4 || Exercises || 450 | ||
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| 12.15 || Exercises || 467 | | 12.15 || Exercises || 467 | ||
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| 12.16 || The vector space | | 12.16 || The vector space \(V_N(C)\) of n-tuples of complex numbers || 468 | ||
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| 12.17 || Exercises || 470 | | 12.17 || Exercises || 470 | ||
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[[Category:Mathematics]] | [[Category:Mathematics]] | ||