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| {{InfoboxBook | | {{InfoboxBook |
| |title=Calculus | | |title=Calculus |
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| |isbn13=978-0471000051 | | |isbn13=978-0471000051 |
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| | The textbook '''''Calculus''''' by [https://en.wikipedia.org/wiki/Tom_M._Apostol Tom Apostol] introduces calculus. Β |
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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. | |
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| == Table of Contents == | | == Table of Contents == |
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| | <nowiki>*</nowiki>I 1.4 || Exercises || 8 | | | <nowiki>*</nowiki>I 1.4 || Exercises || 8 |
| |- | | |- |
| | I 1.5 || A critical analysis of the Archimedes' method || 8 | | | I 1.5 || Rational numbers || 8 |
| |- | | |- |
| | I 1.6 || The approach to calculus to be used in this book || 10 | | | I 1.6 || Multiplicative inverses || 10 |
| |- Β | | |- Β |
| ! colspan="3" | Part 2: Some Basic Concepts of the Theory of Sets | | ! colspan="3" | Chapter 2: Linear Equations |
| |-
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| | I 2.1 || Introduction to set theory || 11
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| | I 2.2 || Notations for designating sets || 12
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| | I 2.3 || Subsets || 12
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| | I 2.4 || Unions, intersections, complements || 13
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| | I 2.5 || Exercises || 15
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| ! colspan="3" | Part 3: A set of Axioms for the Real-Number System
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| | I 3.1 || Introduction || 17
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| | I 3.2 || The field axioms || 17
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| | <nowiki>*</nowiki>I 3.3 || Exercises || 19
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| | I 3.4 || The order axioms || 19
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| | <nowiki>*</nowiki>I 3.5 || Exercises || 21
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| | I 3.6 || Integers and rational numbers || 21
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| | I 3.7 || Geometric interpretation of real numbers as points on a line || 22
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| | I 3.8 || Upper bound of a set, maximum element, least upper bound (supremum) || 23
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| | I 3.9 || The least-Upper-bound axiom (completeness axiom) || 25
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| | I 3.10 || The Archimedean property of the real-number system || 25
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| | I 3.11 || Fundamental properties of the supremum and infimum || 26
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| | <nowiki>*</nowiki>I 3.12 || Exercises || 28
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| | <nowiki>*</nowiki>I 3.13 || Existence of square roots of nonnegative real numbers || 29
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| | <nowiki>*</nowiki>I 3.14 || Roots of higher order. Rational powers || 30
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| | <nowiki>*</nowiki>I 3.15 || Representation of real numbers by decimals || 30
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| |-
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| ! colspan="3" | Part 4: Mathematical Induction, Summation Notation, and Related Topics
| |
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| | I 4.1 || An example of a proof by mathematical induction || 32
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| | I 4.2 || The principle of mathematical induction || 34
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| | <nowiki>*</nowiki>I 4.3 || The well-ordering principle || 34
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| | I 4.4 || Exercises || 35
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| | <nowiki>*</nowiki>I 4.5 || Proof of the well-ordering principle || 37
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| | I 4.6 || The summation notation || 37
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| | I 4.7 || Exercises || 39
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| | I 4.8 || Absolute values and the triangle inequality || 41
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| | I 4.9 || Exercises || 43
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| | <nowiki>*</nowiki>I 4.10 || Miscellaneous exercises involving induction || 44
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| ! colspan="3" | 1. THE CONCEPTS OF INTEGRAL CALCULUS
| |
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| | 1.1 || The basic ideas of Cartesian geometry || 48
| |
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| | 1.2 || Functions. Informal description and examples || 50
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| | <nowiki>*</nowiki>1.3 || Functions. Formal definition as a set of ordered pairs || 53
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| |-
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| | 1.4 || More examples of real functions || 54
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| |-
| |
| | 1.5 || Exercises || 56
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| | 1.6 || The concept of area as a set function || 57
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| | 1.7 || Exercises || 60
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| | 1.8 || Intervals and ordinate sets || 60
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| | 1.9 || Partitions and step functions || 61
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| | 1.10 || Sum and product of step functions || 63
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| | 1.11 || Exercises || 63
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| | 1.12 || The definition of the integral for step functions || 64
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| | 1.13 || Properties of the integral of a step function || 66
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| |-
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| | 1.14 || Other notations for integrals || 69
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| | 1.15 || Exercises || 70
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| | 1.16 || The integral of more general functions || 72
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| |-
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| | 1.17 || Upper and lower integrals || 74
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| | 1.18 || The area of an ordinate set expressed as an integral || 75
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| | 1.19 || Informal remarks on the theory and technique of integration || 75
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| | 1.20 || Monotonic and piecewise monotonic functions. Definitions and examples || 76
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| | 1.21 || Integrability of bounded monotonic functions || 77
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| | 1.22 || Calculation of the integral of a bounded monotonic function || 79
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| | 1.23 || Calculation of the integral <math>\int_0^b x^p dx</math> when <math>p</math> is a positive integer || 79
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| | 1.24 || The basic properties of the integral || 80
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| | 1.25 || Integration of polynomials || 81
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| | 1.26 || Exercises || 83
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| | 1.27 || Proofs of the basic properties of the integral || 84
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| ! colspan="3" | 2. SOME APPLICATIONS OF INTEGRATION
| |
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| | 2.1 || Introduction || 88
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| | 2.2 || The area of a region between two graphs expressed as an integral || 88
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| | 2.3 || Worked examples || 89
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| | 2.4 || Exercises || 94
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| | 2.5 || The trigonometric functions || 94
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| | 2.6 || Integration formulas for the sine and cosine || 94
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| | 2.7 || A geometric description of the sine and cosine functions || 94
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| | 2.8 || Exercises || 94
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| | 2.9 || Polar coordinates || 94
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| | 2.10 || The integral for area in polar coordinates || 94
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| | 2.11 || Exercises || 94
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| | 2.12 || Application of integration to the calculation of volume || 94
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| | 2.13 || Exercises || 94
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| | 2.14 || Application of integration to the calculation of work || 94
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| | 2.15 || Exercises || 94
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| | 2.16 || Average value of a function || 94
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| | 2.17 || Exercises || 94
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| | 2.18 || The integral as a function of the upper limit. Indefinite integrals || 94
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| | 2.19 || Exercises || 94
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| ! colspan="3" | 3. CONTINUOUS FUNCTIONS
| |
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| | 3.1 || Informal description of continuity || 126
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| | 3.2 || The definition of the limit of a function || 127
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| | 3.3 || The definition of continuity of a function || 130
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| | 3.4 || The basic limit theorems. More examples of continuous functions || 131
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| | 3.5 || Proofs of the basic limit theorems || 135
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| | 3.6 || Exercises || 138
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| | 3.7 || Composite functions and continuity || 140
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| | 3.8 || Exercises || 142
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| | 3.9 || Bolzano's theorem for continuous functions || 142
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| | 3.10 || The intermediate-value theorem for continuous functions || 144
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| | 3.11 || Exercises || 145
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| | 3.12 || The process of inversion || 146
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| | 3.13 || Properties of functions preserved by inversion || 147
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| | 3.14 || Inverses of piecewise monotonic functions || 148
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| | 3.15 || Exercises || 149
| |
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| | 3.16 || The extreme-value theorem for continuous functions || 150
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| | 3.17 || The small-span theorem for continuous functions (uniform continuity) || 152
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| | 3.18 || The integrability theorem for continuous functions || 152
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| | 3.19 || Mean-value theorems for integrals of continuous functions || 154
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| | 3.20 || Exercises || 155
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| ! colspan="3" | 4. DIFFERENTIAL CALCULUS
| |
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| | 4.1 || Historical introduction || 156
| |
| |-
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| | 4.2 || A problem involving velocity || 157
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| | 4.3 || The derivative of a function || 159
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| | 4.4 || Examples of derivatives || 161
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| | 4.5 || The algebra of derivatives || 164
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| | 4.6 || Exercises || 167
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| | 4.7 || Geometric interpretation of the derivative as a slope || 169
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| | 4.8 || Other notations for derivatives || 171
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| | 4.9 || Exercises || 173
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| | 4.10 || The chain rule for differentiating composite functions || 174
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| | 4.11 || Applications of the chain rule. Related rates and implicit differentiation || 176
| |
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| | 4.12 || Exercises || 179
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| | 4.13 || Applications of the differentiation to extreme values of cuntions|| 181
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| | 4.14 || The mean-value theorem for derivatives || 183
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| | 4.15 || Exercises || 186
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| | 4.16 || Applications of the mean-value theorem to geometric properties of functions || 187
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| | 4.17 || Second-derivative test for extrema || 188
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| | 4.18 || Curve sketching || 189
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| | 4.19 || Exercises || 191
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| | 4.20 || Worked examples of extremum problems || 191
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| | 4.21 || Exercises || 194
| |
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| | <nowiki>*</nowiki>4.22 || Partial derivatives || 196
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| | <nowiki>*</nowiki>4.23 || Exercises || 201
| |
| |-
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| ! colspan="3" | 5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION
| |
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| | 5.1 || The derivative of an indefinite integral. The first fundamental theorem of calculus || 202
| |
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| | 5.2 || The zero-derivative theorem || 204
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| | 5.3 || Primitive functions and the second fundamental theorem of calculus || 205
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| | 5.4 || Properties of a function deduced from properties of its derivative || 207
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| |-
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| | 5.5 || Exercises || 208
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| | 5.6 || The Leibniz notation for primitives || 210
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| | 5.7 || Integration by substitution || 212
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| | 5.8 || Exercises || 216
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| | 5.9 || Integration by parts || 217
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| | 5.10 || Exercises || 220
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| | <nowiki>*</nowiki>5.11 || Miscellaneous review exercises || 222
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| ! colspan="3" | 6. THE LOGARITHM, THE EXPONENTIAL, AND THE INVERSE TRIGONOMETRIC FUNCTIONS
| |
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| | 6.1 || Introduction || 226
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| | 6.2 || Motivation for the definition of the natural logarithm as an integral || 227
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| | 6.3 || The definition of the logarithm. Basic properties || 229
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| | 6.4 || The graph of the natural logarithm || 230
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| | 6.5 || Consequences of the functional equation <math>L(ab) = L(a) + L(b)</math> || 230
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| | 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
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| | 6.8 || Logarithmic differentiation || 235
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| | 6.9 || Exercises || 236
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| | 6.10 || Polynomial approximations to the logarithm || 236
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| | 6.11 || Exercises || 242
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| | 6.12 || The exponential function || 242
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| | 6.13 || Exponentials expressed as powers of e || 242
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| | 6.14 || The definition of <math>e^x</math> for arbitrary real x || 244
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| | 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
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| | 6.17 || Exercises || 248
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| | 6.18 || The hyperbolic functions || 251
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| | 6.19 || Exercises || 251
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| | 6.20 || Derivatives of inverse functions || 252
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| | 6.21 || Inverses of the trigonometric functions || 253
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| | 6.22 || Exercises || 256
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| | 6.23 || Integration by partial fractions || 258
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| | 6.24 || Integrals which can be transformed into integrals of rational functions || 264
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| | 6.25 || Exercises || 267
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| | 6.26 || Miscellaneous review exercises || 268
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| ! colspan="3" | 7. POLYNOMIAL APPROXIMATIONS TO FUNCTIONS
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| | 7.1 || Introduction || 272
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| | 7.2 || The Taylor polynomials generated by a function || 273
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| | 7.3 || Calculus of Taylor polynomials || 275
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| | 7.4 || Exercises || 278
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| | 7.5 || Taylor's formula with remainder || 278
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| | 7.6 || Estimates for the error in Taylor's formula || 280
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| | <nowiki>*</nowiki>7.7 || Other forms of the remainder in Taylor's formula || 283
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| | 7.8 || Exercises || 284
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| | 7.9 || Further remarks on the error in Taylor's formula. The o-notation || 286
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| | 7.10 || Applications to indeterminate forms || 289
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| | 7.11 || Exercises || 290
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| | 7.12 || L'Hopital's rule for the indeterminate form 0/0 || 292
| |
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| | 7.13 || Exercises || 295
| |
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| | 7.14 || The symbols <math>+\inf</math> and <math>-\inf</math>. Extension of L'Hopital's rule || 296
| |
| |-
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| | 7.15 || Infinite limits || 298
| |
| |-
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| | 7.16 || The behavior of log<math>x</math> and <math>e^x</math> for large <math>x</math> || 300
| |
| |-
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| | 7.17 || Exercises || 303
| |
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| ! colspan="3" | 8. INTRODUCTION TO DIFFERENTIAL EQUATIONS
| |
| |-
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| | 8.1 || Introduction || 305
| |
| |-
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| | 8.2 || Terminology and notation || 306
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| | 8.3 || A first-order differential equation for the exponential function || 307
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| | 8.4 || First-order linear differential equations || 308
| |
| |- | | |- |
| | 8.5 || Exercises || 311 | | | 1 || Equations in two unknowns || 53 |
| |- | | |- |
| | 8.6 || Some physical problems leading to first-order linear differential equations || 313 | | | 2 || Equations in three unknowns || 57 |
| |- | | |- |
| | 8.7 || Exercises || 319 | | ! colspan="3" | Chapter 3: Real Numbers |
| |- | | |- |
| | 8.8 || Linear equations of second order with constant coefficients || 322 | | | 1 || Addition and multiplication || 61 |
| |- | | |- |
| | 8.9 || Existence of solutions of the equation <math>y^{''} + by = 0</math> || 323 | | | 2 || Real numbers: positivity || 64 |
| |- | | |- |
| | 8.10 || Reduction of the general equation to the special case <math>y^{''} + by = 0</math> || 324 | | | 3 || Powers and roots || 70 |
| |- | | |- |
| | 8.11 || Uniqueness theorem for the equation <math>y^{''} + by = 0</math> || 324 | | | 4 || Inequalities || 75 |
| |- | | |- |
| | 8.12 || Complete solution of the equation <math>y^{''} + by = 0</math> || 326
| | ! colspan="3" | Chapter 4: Quadratic Equations |
| |- | | |- |
| | 8.13 || Complete solution of the equation <math>y^{''} + ay^{'} + by = 0</math> || 326
| | ! colspan="3" | Interlude: On Logic and Mathematical Expressions |
| |- | | |- |
| | 8.14 || Exercises || 328 | | | 1 || On reading books || 93 |
| |- | | |- |
| | 8.15 || Nonhomogeneous linear equations of second order with constant coefficients || 329 | | | 2 || Logic || 94 |
| |- | | |- |
| | 8.16 || Special methods for determining a particular solution of the nonhomogeneous equation <math>y^{''} + ay^{'} + by = R</math> || 332 | | | 3 || Sets and elements || 99 |
| |- | | |- |
| | 8.17 || Exercises || 333 | | | 4 || Notation || 100 |
| |- | | |- |
| | 8.18 || Examples of physical problems leading to linear second-order equations with constant coefficients || 334 | | ! colspan="3" | PART II: INTUITIVE GEOMETRY |
| |- | | |- |
| | 8.19 || Exercises || 339 | | ! colspan="3" | Chapter 5: Distance and Angles |
| |- | | |- |
| | 8.20 || Remarks concerning nonlinear differential equations || 339 | | | 1 || Distance || 107 |
| |- | | |- |
| | 8.21 || Integral curves and direction fields || 341 | | | 2 || Angles || 110 |
| |- | | |- |
| | 8.22 || Exercises || 344 | | | 3 || The Pythagoras theorem || 120 |
| |- | | |- |
| | 8.23 || First-order separable equations || 345 | | ! colspan="3" | Chapter 6: Isometries |
| |- | | |- |
| | 8.24 || Exercises || 347 | | | 1 || Some standard mappings of the plane || 133 |
| |- | | |- |
| | 8.25 || Homogeneous first-order equations || 347 | | | 2 || Isometries || 143 |
| |- | | |- |
| | 8.26 || Exercises || 350 | | | 3 || Composition of isometries || 150 |
| |- | | |- |
| | 8.27 || Some geometrical and physical problems leading to first-order equations || 351 | | | 4 || Inverse of isometries || 155 |
| |- | | |- |
| | 8.28 || Miscellaneous review exercises || 355 | | | 5 || Characterization of isometries || 163 |
| |- | | |- |
| ! colspan="3" | 9. COMPLEX NUMBERS
| | | 6 || Congruences || 166 |
| |- | | |- |
| | 9.1 || Historical introduction || 358 | | ! colspan="3" | Chapter 7: Area and Applications |
| |- | | |- |
| | 9.2 || Definitions and field properties || 358 | | | 1 || Area of a disc of radius ''r'' || 173 |
| |- | | |- |
| | 9.3 || The complex numbers as an extension of the real numbers || 360 | | | 2 || Circumference of a circle of radius ''r'' || 180 |
| |- | | |- |
| | 9.4 || The imaginary unit <math>i</math> || 361 | | ! colspan="3" | PART III: COORDINATE GEOMETRY |
| |- | | |- |
| | 9.5 || Geometric interpretation. Modulus and argument || 362 | | ! colspan="3" | Chapter 8: Coordinates and Geometry |
| |- | | |- |
| | 9.6 || Exercises || 365 | | | 1 || Coordinate systems || 191 |
| |- | | |- |
| | 9.7 || Complex exponentials || 366 | | | 2 || Distance between points || 197 |
| |- | | |- |
| | 9.8 || Complex-valued functions || 368 | | | 3 || Equation of a circle || 203 |
| |- | | |- |
| | 9.9 || Examples of differentiation and integration formulas || 369 | | | 4 || Rational points on a circle || 206 |
| |- | | |- |
| | 9.10 || Exercises || 371 | | ! colspan="3" | Chapter 9: Operations on Points |
| |- | | |- |
| ! colspan="3" | 10. SEQUENCES, INFINITE SERIES, IMPROPER INTEGRALS
| | | 1 || Dilations and reflections || 213 |
| |- | | |- |
| | 10.1 || Zeno's paradox || 374 | | | 2 || Addition, subtraction, and the parallelogram law || 218 |
| |- | | |- |
| | 10.2 || Sequences || 378 | | ! colspan="3" | Chapter 10: Segments, Rays, and Lines |
| |- | | |- |
| | 10.3 || Monotonic sequences of real numbers || 381 | | | 1 || Segments || 229 |
| |- | | |- |
| | 10.4 || Exercises || 382 | | | 2 || Rays || 231 |
| |- | | |- |
| | 10.5 || Infinite series || 383 | | | 3 || Lines || 236 |
| |- | | |- |
| | 10.6 || The linearity property of convergent series || 385 | | | 4 || Ordinary equation for a line || 246 |
| |- | | |- |
| | 10.7 || Telescoping series || 386 | | ! colspan="3" | Chapter 11: Trigonometry |
| |- | | |- |
| | 10.8 || The geometric series || 388 | | | 1 || Radian measure || 249 |
| |- | | |- |
| | 10.9 || Exercises || 391 | | | 2 || Sine and cosine || 252 |
| |- | | |- |
| | <nowiki>*</nowiki>10.10 || Exercises on decimal expansions || 393 | | | 3 || The graphs || 264 |
| |- | | |- |
| | 10.11 || Tests for convergence || 394 | | | 4 || The tangent || 266 |
| |- | | |- |
| | 10.12 || Comparison tests for series of nonnegative terms || 394 | | | 5 || Addition formulas || 272 |
| |- | | |- |
| | 10.13 || The integral test || 397 | | | 6 || Rotations || 277 |
| |- | | |- |
| | 10.14 || Exercises || 398 | | ! colspan="3" | Chapter 12: Some Analytic Geometry |
| |- | | |- |
| | 10.15 || The root test and the ratio test for series of nonnegative terms || 399 | | | 1 || The straight line again || 281 |
| |- | | |- |
| | 10.16 || Exercises || 402 | | | 2 || The parabola || 291 |
| |- | | |- |
| | 10.17 || Alternating series || 403 | | | 3 || The ellipse || 297 |
| |- | | |- |
| | 10.18 || Conditional and absolute convergence || 406 | | | 4 || The hyperbola || 300 |
| |- | | |- |
| | 10.19 || The convergence tests of Dirichlet and Abel || 407 | | | 5 || Rotation of hyperbolas || 305 |
| |- | | |- |
| | 10.20 || Exercises || 409 | | ! colspan="3" | PART IV: MISCELLANEOUS |
| |- | | |- |
| | <nowiki>*</nowiki>10.21 || Rearrangements of series || 411 | | ! colspan="3" | Chapter 13: Functions |
| |- | | |- |
| | 10.22 || Miscellaneous review exercises || 414 | | | 1 || Definition of a function || 313 |
| |- | | |- |
| | 10.23 || Improper integrals || 416 | | | 2 || Polynomial functions || 318 |
| |- | | |- |
| | 10.24 || Exercises || 420 | | | 3 || Graphs of functions || 330 |
| |- | | |- |
| ! colspan="3" | 11. SEQUENCES AND SERIES OF FUNCTIONS
| | | 4 || Exponential function || 333 |
| |- | | |- |
| | 11.1 || Pointwise convergence of sequences of functions || 422 | | | 5 || Logarithms || 338 |
| |- | | |- |
| | 11.2 || Uniform convergence of sequences of functions || 423 | | ! colspan="3" | Chapter 14: Mappings |
| |- | | |- |
| | 11.3 || Uniform convergence and continuity || 424 | | | 1 || Definition || 345 |
| |- | | |- |
| | 11.4 || Uniform convergence and integration || 425 | | | 2 || Formalism of mappings || 351 |
| |- | | |- |
| | 11.5 || A sufficient condition for uniform convergence || 427 | | | 3 || Permutations || 359 |
| |- | | |- |
| | 11.6 || Power series. Circle of convergence || 428 | | ! colspan="3" | Chapter 15: Complex Numbers |
| |- | | |- |
| | 11.7 || Exercises || 430 | | | 1 || The complex plane || 375 |
| |- | | |- |
| | 11.8 || Properties of functions represented by real power series || 431 | | | 2 || Polar form || 380 |
| |- | | |- |
| | 11.9 || The Taylor's series generated by a function || 434 | | ! colspan="3" | Chapter 16: Induction and Summations |
| |- | | |- |
| | 11.10 || A sufficient condition for convergence of a Taylor's series || 435 | | | 1 || Induction || 383 |
| |- | | |- |
| | 11.11 || Power-series expansions for the exponential and trigonometric functions || 435 | | | 2 || Summations || 388 |
| |- | | |- |
| | <nowiki>*</nowiki>11.12 || Bernstein's theorem || 437 | | | 3 || Geometric series || 396 |
| |- | | |- |
| | 11.13 || Exercises || 438 | | ! colspan="3" | Chapter 17: Determinants |
| |- | | |- |
| | 11.14 || Power series and differential equations || 439 | | | 1 || Matrices || 401 |
| |- | | |- |
| | 11.15 || The binomial series || 441 | | | 2 || Determinants of order 2 || 406 |
| |- | | |- |
| | 11.16 || Exercises || 443 | | | 3 || Properties of 2 x 2 determinants || 409 |
| |- | | |- |
| ! colspan="3" | 12. VECTOR ALGEBRA
| | | 4 || Determinants of order 3 || 414 |
| |- | | |- |
| | 12.1 || Historical introduction || 445 | | | 5 || Properties of 3 x 3 determinants || 418 |
| |- | | |- |
| | 12.2 || The vector space of n-tuples of real numbers || 446 | | | 6 || Cramer's Rule || 424 |
| |- | | |- |
| | 12.3 || Geometric interpretation for <math>n \leq 3</math> || 448
| | ! colspan="2" | Index || 429 |
| |-
| |
| | 12.4 || Exercises || 450
| |
| |-
| |
| | 12.5 || The dot product || 451
| |
| |-
| |
| | 12.6 || Length or norm of a vector|| 453
| |
| |-
| |
| | 12.7 || Orthogonality of vectors || 455
| |
| |-
| |
| | 12.8 || Exercises || 456
| |
| |-
| |
| | 12.9 || Projections. Angle between vectors in n-space || 457
| |
| |-
| |
| | 12.10 || The unit coordinate vectors || 458
| |
| |-
| |
| | 12.11 || Exercises || 460
| |
| |-
| |
| | 12.12 || The linear span of a finite set of vectors || 462
| |
| |-
| |
| | 12.13 || Linear independence || 463
| |
| |-
| |
| | 12.14 || Bases || 466
| |
| |-
| |
| | 12.15 || Exercises || 467
| |
| |-
| |
| | 12.16 || The vector space <math>V_N(C)</math> of n-tuples of complex numbers || 468
| |
| |-
| |
| | 12.17 || Exercises || 470
| |
| |-
| |
| ! colspan="3" | 13. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY
| |
| |-
| |
| | 13.1 || Introduction || 471
| |
| |-
| |
| | 13.2 || Lines in n-space || 472
| |
| |-
| |
| | 13.3 || Some simple properties of straight lines || 473
| |
| |-
| |
| | 13.4 || Lines and vector-valued functions || 474
| |
| |-
| |
| | 13.5 || Exercises || 477
| |
| |-
| |
| | 13.6 || Planes in Euclidean n-space || 478
| |
| |-
| |
| | 13.7 || Planes and vector-valued functions || 481
| |
| |-
| |
| | 13.8 || Exercises || 482
| |
| |-
| |
| | 13.9 || The cross product || 483
| |
| |-
| |
| | 13.10 || The cross product expressed as a determinant || 486
| |
| |-
| |
| | 13.11 || Exercises || 487
| |
| |-
| |
| | 13.12 || The scalar triple product || 488
| |
| |-
| |
| | 13.13 || Cramer's rule for solving a system of three linear equations || 490
| |
| |-
| |
| | 13.14 || Exercises || 491
| |
| |-
| |
| | 13.15 || Normal vectors to planes || 493
| |
| |-
| |
| | 13.16 || Linear Cartesian equations for planes || 494
| |
| |-
| |
| | 13.17 || Exercises || 496
| |
| |-
| |
| | 13.18 || The conic sections || 497
| |
| |-
| |
| | 13.19 || Eccentricity of conic sections || 500
| |
| |-
| |
| | 13.20 || Polar equations for conic sections || 501
| |
| |-
| |
| | 13.21 || Exercises || 503
| |
| |-
| |
| | 13.22 || Conic sections symmetric about the origin || 504
| |
| |-
| |
| | 13.23 || Cartesian equations for the conic sections || 505
| |
| |-
| |
| | 13.24 || Exercises || 508
| |
| |-
| |
| | 13.25 || Miscellaneous exercises on conic sections || 509
| |
| |-
| |
| ! colspan="3" | 14. CALCULUS OF VECTOR-VALUED FUNCTIONS
| |
| |-
| |
| | 14.1 || Vector-valued functions of a real variable || 512
| |
| |-
| |
| | 14.2 || Algebraic operations. Components || 512
| |
| |-
| |
| | 14.3 || Limits, derivatives, and integrals || 513
| |
| |-
| |
| | 14.4 || Exercises || 516
| |
| |-
| |
| | 14.5 || Applications to curves. Tangency || 517
| |
| |-
| |
| | 14.6 || Applications to curvilinear motion. Velocity, speed, and acceleration || 520
| |
| |-
| |
| | 14.7 || Exercises || 524
| |
| |-
| |
| | 14.8 || The unit tangent, the principal normal, and the osculating plane of a curve || 525
| |
| |-
| |
| | 14.9 || Exercises || 528
| |
| |-
| |
| | 14.10 || The definition of arc length || 529
| |
| |-
| |
| | 14.11 || Additivity of arc length || 532
| |
| |-
| |
| | 14.12 || The arc-length function || 533
| |
| |-
| |
| | 14.13 || Exercises || 535
| |
| |-
| |
| | 14.14 || Curvature of a curve || 536
| |
| |-
| |
| | 14.15 || Exercises || 538
| |
| |-
| |
| | 14.16 || Velocity and acceleration in polar coordinates || 540
| |
| |-
| |
| | 14.17 || Plane motion with radial acceleration || 542
| |
| |-
| |
| | 14.18 || Cylindrical coordinates || 543
| |
| |-
| |
| | 14.19 || Exercises || 543
| |
| |-
| |
| | 14.20 || Applications to planetary motion || 545
| |
| |-
| |
| | 14.21 || Miscellaneous review exercises || 549
| |
| |-
| |
| ! colspan="3" | 15. LINEAR SPACES
| |
| |-
| |
| | 15.1 || Introduction || 551
| |
| |-
| |
| | 15.2 || The definition of a linear space || 551
| |
| |-
| |
| | 15.3 || Examples of linear spaces || 552
| |
| |-
| |
| | 15.4 || Elementary consequences of the axioms || 554
| |
| |-
| |
| | 15.5 || Exercises || 555
| |
| |-
| |
| | 15.6 || Subspaces of a linear space || 556
| |
| |-
| |
| | 15.7 || Dependent and independent sets in a linear space || 557
| |
| |-
| |
| | 15.8 || Bases and dimension || 559
| |
| |-
| |
| | 15.9 || Exercises || 560
| |
| |-
| |
| | 15.10 || Inner products, Euclidean spaces, norms || 561
| |
| |-
| |
| | 15.11 || Orthogonality in a Euclidean space || 564
| |
| |-
| |
| | 15.12 || Exercises || 566
| |
| |-
| |
| | 15.13 || Construction of orthogonal sets. The Gram-Schmidt process || 568
| |
| |-
| |
| | 15.14 || Orthogonal complements. Projections || 572
| |
| |-
| |
| | 15.15 || Best approximation of elements in a Euclidean space by elements in a finite-dimensional subspace || 574
| |
| |-
| |
| | 15.16 || Exercises || 576
| |
| |-
| |
| ! colspan="3" | 16. LINEAR TRANSFORMATIONS AND MATRICES
| |
| |-
| |
| | 16.1 || Linear transformations || 578
| |
| |-
| |
| | 16.2 || Null space and range || 579
| |
| |-
| |
| | 16.3 || Nullity and rank || 581
| |
| |-
| |
| | 16.4 || Exercises || 582
| |
| |-
| |
| | 16.5 || Algebraic operations on linear transformations || 583
| |
| |-
| |
| | 16.6 || Inverses || 585
| |
| |-
| |
| | 16.7 || One-to-one linear transformations || 587
| |
| |-
| |
| | 16.8 || Exercises || 589
| |
| |-
| |
| | 16.9 || Linear transformations with prescribed values || 590
| |
| |-
| |
| | 16.10 || Matrix representations of linear transformations || 591
| |
| |-
| |
| | 16.11 || Construction of a matrix representation in diagonal form || 594
| |
| |-
| |
| | 16.12 || Exercises || 596
| |
| |-
| |
| | 16.13 || Linear spaces of matrices || 597
| |
| |-
| |
| | 16.14 || Isomorphism between linear transformations and matrices || 599
| |
| |-
| |
| | 16.15 || Multiplication of matrices || 600
| |
| |-
| |
| | 16.16 || Exercises || 603
| |
| |-
| |
| | 16.17 || Systems of linear equations || 605
| |
| |-
| |
| | 16.18 || Computation techniques || 607
| |
| |-
| |
| | 16.19 || Inverses of square matrices || 611
| |
| |-
| |
| | 16.20 || Exercises || 613
| |
| |-
| |
| | 16.21 || Miscellaneous exercises on matrices || 614
| |
| |-
| |
| ! colspan="2" | Answers to exercises || 617
| |
| |-
| |
| ! colspan="2" | Index || 657 | |
| |- | | |- |
| |} | | |} |
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| [[Category:Mathematics]] | | [[Category:Mathematics]] |
| {{Stub}}
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