Advanced calculus for economics and finance: theory and methods
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham
Springer
[2023]
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| Series: | Classroom companion: Economics
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xi, 315 Seiten Diagramme |
| ISBN: | 9783031303159 |
| ISSN: | 2662-2890 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Contents 1 Preliminaries .................................................................................................. 1.1 Sets, Equivalence Relations, and Functions...................................... 1.2 Order Relations, Supremum, and Infimum ...................................... 1.3 Countable Sets ......................................................................... 1.4 Operations and Fields ........................................................................ 1.5 Real Numbers ...................................................................................... 1.6 Convexity and Concavity ................................................................... Exercises ....................................................................................................... 2 Topology ........................................................................................................... Definition and Basic Properties ........................................................ Base of a Topology ............................................................................ 2.2.1 Countability .............................................................................. 2.2.2 Euclidean Topology ................................................................. 2.3 Cover and Compactness ..................................................................... 2.3.1 Hausdorff Spaces .................................................................... 2.3.2 Compactness in Euclidean Topology ..................................... 2.3.3 The Extended Real Number System
...................................... 2.4 Connectedness .................................................................................... 2.5 Limit of Functions and Continuity ................................................... 2.5.1 Continuity in Euclidean Topology ........................................ Exercises ....................................................................................................... 2.1 2.2 3 Metric Spaces .................................................................................................. 3.1 Definition and Basic Properties ........................................................ 3.2 Metric and Topology ......................................................................... 3.3 Uniform Continuity ........................................................................... 3.4 Lipschitz Continuity ........................................................................... Exercises ....................................................................................................... 4 Normed Spaces ............................................................................................... 4.1 4.2 Definition and Basic Properties ......................................................... Example of Normed Spaces ............................................................. 4.2.1 Euclidean Norm in B ........................................................... 1 1 4 6 7 9 18 21 25 25 29 31 32 34 35 38 40 40 42 47 50 55 55 57 59 60 61 65 65 67 67 ix
Contents x 4.2.2 p-Norm in R” ......................................................................... 4.2.3 Operator Norm ......................................................................... 4.3 Finite-Dimensional Normed Spaces .................................................. 4.3.1 Equivalence of Norms in R ................................................. Exercises ....................................................................................................... 5 Sequences and Series ................... Sequences in Topological Spaces ...................................................... 5.1.1 Subsequences .......................................................................... 5.1.2 Sequences and Functions ...................................................... 5.1.3 Uniqueness of Limit............................................................... 5.2 Sequences in Metric Spaces .............................................. 5.2.1 Cauchy Sequences and Complete Spaces ........................... 5.3 Sequences in R..................................................................................... 5.3.1 Upper and Lower Limits ....................................................... 5.3.2 Infinity and Infinitesimals ..................................................... 5.4 Sequences in Normed Spaces ............................................................. 5.5 Series in R .................................................. i....................................... 5.5.1 Series with Decreasing Terms ............................................... 5.5.2
Tests Based on the Asymptotic Behaviour of Terms ........... 5.6 Sequences and Series of Functions.................................................... 5.6.1 Uniform Convergence ................................ Exercises .................................. 5.1 69 71 73 76 78 81 81 82 83 84 85 85 91 99 101 105 108 110 112 116 117 119 Limit of Real Functions .................................................................... Continuity of Real Functions ........................................................... Differential Analysis .......................................................................... 6.3.1 Higher-Order Derivatives ........................................................ 6.3.2 Derivatives and Function Behaviour ..................................... 6.3.3 Derivatives and Limits ............ 6.4 Taylor Polynomial and PowerSeries Expansion .............................. Exercises ....................................................................................................... 125 125 127 130 136 136 140 143 151 7 Differential Calculus of Functions of Several Variables ........................ 7.1 Limits and Continuity in R ............................................................... 7.2 Differential Analysis in R” ................................................................ 7.3 Mean Value Theorems ......................................................................... 7.4 Higher-Order Derivatives and Taylor Polynomial ............................ 7.4.1 Local Maxima and Minima .................................................... 7.5 Inverse
Function Theorem .................................................................. 7.6 Implicit Function Theorem ................................................................ 7.6.1 Real Functions of Two Variables .......................................... 7.6.2 Real Functions of Several Variables ........................... 155 155 157 170 171 175 178 181 182 184 6 Differential Calculus of Functions of One Variable ............................... 6.1 6.2 6.3
Contents 8 7.6.3 Vector Functions of Several Variables .................................. 7.6.4 Dependent and Independent Functions ................................ 7.7 Constrained Optimisation ................................................................... 7.7.1 One Dimensional Problems .................................................... 7.7.2 Two Dimensional Problems ................................................. 7.7.3 Theorems of the Alternatives ................................................ 7.7.4 First-Order Conditions .......................................................... 7.7.5 Second-Order Conditions ....................................................... 7.7.6 Envelope Theorem ................................................................. Ί.Ί.Ί Minimisation Problems .......................................................... Exercises ....................................................................................................... 186 188 189 190 193 195 197 202 208 211 211 Integral Calculus ........................................................................................... 215 215 219 221 223 225 226 232 234 239 241 8.1 Definite Integrals .................................................................................. 8.1.1 Properties of the Definite Integral ......................................... 8.1.2 Riemann IntegrableFunctions ................................................ 8.1.3 Improper Integrals................................................................... 8.1.4 Integral of Vector-Valued
Functions..................................... 8.2 The Fundamental Theorem ofCalculus ............................................ 8.3 Riemann-Stieltjes Integral ................................................................. 8.3.1 Stieltjes Integrable Functions ................................................ 8.3.2 Properties of the Stieltjes Integral .......................................... Exercises ....................................................................................................... 9 xi Measure Theory ............................................................................................. Algebras, Measurable Spaces, and Measures .................................. 9.1.1 Complete Measure Space ....................................................... 9.1.2 Borel σ-Algebra ..................................................................... 9.1.3 Lebesgue Measure .........;....................................................... 9.2 Measurable Functions ....................................................................... 9.2.1 Measurable Real-Valued Functions ...................................... 9.3 Lebesgue Integral .............................................................................. 9.3.1 Lebesgue and Riemann Integral on R .................................... 9.4 Product Measure Space ...................................................................... 9.4.1 Product σ-Algebra ................................................................. 9.4.2 Product Measure
.................................................................... 9.4.3 Multiple Integrals .................................................................. 9.5 Probability Measure ........................................................................... 9.5.1 Multiple Random Variables ................................................... 9.5.2 Banach Space of Square Summable Random Variables .... Exercises ....................................................................................................... 9.1 245 245 249 250 251 256 259 261 274 277 278 279 283 287 292 294 296 Appendix A: Cauchy Initial Value Problem .................................................. 301 Appendix В : Brouwer Fixed Point Theorem ................................................. 305 Index ........................................................................................................................ 309
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| adam_txt |
Contents 1 Preliminaries . 1.1 Sets, Equivalence Relations, and Functions. 1.2 Order Relations, Supremum, and Infimum . 1.3 Countable Sets . 1.4 Operations and Fields . 1.5 Real Numbers . 1.6 Convexity and Concavity . Exercises . 2 Topology . Definition and Basic Properties . Base of a Topology . 2.2.1 Countability . 2.2.2 Euclidean Topology . 2.3 Cover and Compactness . 2.3.1 Hausdorff Spaces . 2.3.2 Compactness in Euclidean Topology . 2.3.3 The Extended Real Number System
. 2.4 Connectedness . 2.5 Limit of Functions and Continuity . 2.5.1 Continuity in Euclidean Topology . Exercises . 2.1 2.2 3 Metric Spaces . 3.1 Definition and Basic Properties . 3.2 Metric and Topology . 3.3 Uniform Continuity . 3.4 Lipschitz Continuity . Exercises . 4 Normed Spaces . 4.1 4.2 Definition and Basic Properties . Example of Normed Spaces . 4.2.1 Euclidean Norm in B" . 1 1 4 6 7 9 18 21 25 25 29 31 32 34 35 38 40 40 42 47 50 55 55 57 59 60 61 65 65 67 67 ix
Contents x 4.2.2 p-Norm in R” . 4.2.3 Operator Norm . 4.3 Finite-Dimensional Normed Spaces . 4.3.1 Equivalence of Norms in R" . Exercises . 5 Sequences and Series . Sequences in Topological Spaces . 5.1.1 Subsequences . 5.1.2 Sequences and Functions . 5.1.3 Uniqueness of Limit. 5.2 Sequences in Metric Spaces . 5.2.1 Cauchy Sequences and Complete Spaces . 5.3 Sequences in R. 5.3.1 Upper and Lower Limits . 5.3.2 Infinity and Infinitesimals . 5.4 Sequences in Normed Spaces . 5.5 Series in R . i. 5.5.1 Series with Decreasing Terms . 5.5.2
Tests Based on the Asymptotic Behaviour of Terms . 5.6 Sequences and Series of Functions. 5.6.1 Uniform Convergence . Exercises . 5.1 69 71 73 76 78 81 81 82 83 84 85 85 91 99 101 105 108 110 112 116 117 119 Limit of Real Functions . Continuity of Real Functions . Differential Analysis . 6.3.1 Higher-Order Derivatives . 6.3.2 Derivatives and Function Behaviour . 6.3.3 Derivatives and Limits . 6.4 Taylor Polynomial and PowerSeries Expansion . Exercises . 125 125 127 130 136 136 140 143 151 7 Differential Calculus of Functions of Several Variables . 7.1 Limits and Continuity in R" . 7.2 Differential Analysis in R” . 7.3 Mean Value Theorems . 7.4 Higher-Order Derivatives and Taylor Polynomial . 7.4.1 Local Maxima and Minima . 7.5 Inverse
Function Theorem . 7.6 Implicit Function Theorem . 7.6.1 Real Functions of Two Variables . 7.6.2 Real Functions of Several Variables . 155 155 157 170 171 175 178 181 182 184 6 Differential Calculus of Functions of One Variable . 6.1 6.2 6.3
Contents 8 7.6.3 Vector Functions of Several Variables . 7.6.4 Dependent and Independent Functions . 7.7 Constrained Optimisation . 7.7.1 One Dimensional Problems . 7.7.2 Two Dimensional Problems . 7.7.3 Theorems of the Alternatives . 7.7.4 First-Order Conditions . 7.7.5 Second-Order Conditions . 7.7.6 Envelope Theorem . Ί.Ί.Ί Minimisation Problems . Exercises . 186 188 189 190 193 195 197 202 208 211 211 Integral Calculus . 215 215 219 221 223 225 226 232 234 239 241 8.1 Definite Integrals . 8.1.1 Properties of the Definite Integral . 8.1.2 Riemann IntegrableFunctions . 8.1.3 Improper Integrals. 8.1.4 Integral of Vector-Valued
Functions. 8.2 The Fundamental Theorem ofCalculus . 8.3 Riemann-Stieltjes Integral . 8.3.1 Stieltjes Integrable Functions . 8.3.2 Properties of the Stieltjes Integral . Exercises . 9 xi Measure Theory . Algebras, Measurable Spaces, and Measures . 9.1.1 Complete Measure Space . 9.1.2 Borel σ-Algebra . 9.1.3 Lebesgue Measure .;. 9.2 Measurable Functions . 9.2.1 Measurable Real-Valued Functions . 9.3 Lebesgue Integral . 9.3.1 Lebesgue and Riemann Integral on R . 9.4 Product Measure Space . 9.4.1 Product σ-Algebra . 9.4.2 Product Measure
. 9.4.3 Multiple Integrals . 9.5 Probability Measure . 9.5.1 Multiple Random Variables . 9.5.2 Banach Space of Square Summable Random Variables . Exercises . 9.1 245 245 249 250 251 256 259 261 274 277 278 279 283 287 292 294 296 Appendix A: Cauchy Initial Value Problem . 301 Appendix В : Brouwer Fixed Point Theorem . 305 Index . 309 |
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| spelling | Bottazzi, Giulio Verfasser (DE-588)171875052 aut Advanced calculus for economics and finance theory and methods Giulio Bottazzi Cham Springer [2023] xi, 315 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Classroom companion: Economics 2662-2890 Quantitative Economics Mathematics in Business, Economics and Finance Statistics in Business, Management, Economics, Finance, Insurance Econometrics Social sciences—Mathematics Statistics Erscheint auch als Online-Ausgabe 978-3-031-30316-6 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034349825&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bottazzi, Giulio Advanced calculus for economics and finance theory and methods Quantitative Economics Mathematics in Business, Economics and Finance Statistics in Business, Management, Economics, Finance, Insurance Econometrics Social sciences—Mathematics Statistics |
| title | Advanced calculus for economics and finance theory and methods |
| title_auth | Advanced calculus for economics and finance theory and methods |
| title_exact_search | Advanced calculus for economics and finance theory and methods |
| title_exact_search_txtP | Advanced calculus for economics and finance theory and methods |
| title_full | Advanced calculus for economics and finance theory and methods Giulio Bottazzi |
| title_fullStr | Advanced calculus for economics and finance theory and methods Giulio Bottazzi |
| title_full_unstemmed | Advanced calculus for economics and finance theory and methods Giulio Bottazzi |
| title_short | Advanced calculus for economics and finance |
| title_sort | advanced calculus for economics and finance theory and methods |
| title_sub | theory and methods |
| topic | Quantitative Economics Mathematics in Business, Economics and Finance Statistics in Business, Management, Economics, Finance, Insurance Econometrics Social sciences—Mathematics Statistics |
| topic_facet | Quantitative Economics Mathematics in Business, Economics and Finance Statistics in Business, Management, Economics, Finance, Insurance Econometrics Social sciences—Mathematics Statistics |
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