Continuous Transformations in Analysis: With an Introduction to Algebraic Topology
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Berlin, Heidelberg
Springer Berlin Heidelberg
1955
|
| Schriftenreihe: | Die Grundlehren der Mathematischen Wissenschaften, In Einzeldarstellungen mit Besonderer Berücksichtigung der Anwendungsgebiete
75 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | The general objective of this treatise is to give a systematic presentation of some of the topological and measure-theoretical foundations of the theory of real-valued functions of several real variables, with particular emphasis upon a line of thought initiated by BANACH, GEOCZE, LEBESGUE, TONELLI, and VITALI. To indicate a basic feature in this line of thought, let us consider a real-valued continuous function I(u) of the single real variable tt. Such a function may be thought of as defining a continuous translormation T under which x = 1 (u) is the image of u. About thirty years ago, BANACH and VITALI observed that the fundamental concepts of bounded variation, absolute continuity, and derivative admit of fruitful geometrical descriptions in terms of the transformation T: x = 1 (u) associated with the function 1 (u). They further noticed that these geometrical descriptions remain meaningful for a continuous transformation T in Euclidean n-space Rff, where T is given by a system of equations of the form 1-/(1 ff) X-I U, . . . ,tt ,. ", and n is an arbitrary positive integer. Accordingly, these geometrical descriptions can be used to define, for continuous transformations in Euclidean n-space Rff, n-dimensional concepts 01 bounded variation and absolute continuity, and to introduce a generalized Jacobian without reference to partial derivatives. These ideas were further developed, generalized, and modified by many mathematicians, and significant applications were made in Calculus of Variations and related fields along the lines initiated by GEOCZE, LEBESGUE, and TONELLI. |
| Beschreibung: | 1 Online-Ressource |
| ISBN: | 9783642859892 9783642859915 |
| DOI: | 10.1007/978-3-642-85989-2 |
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Datensatz im Suchindex
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| building | Verbundindex |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
| dewey-raw | 510 |
| dewey-search | 510 |
| dewey-sort | 3510 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-3-642-85989-2 |
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| isbn | 9783642859892 9783642859915 |
| language | English |
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| series | Die Grundlehren der Mathematischen Wissenschaften, In Einzeldarstellungen mit Besonderer Berücksichtigung der Anwendungsgebiete |
| series2 | Die Grundlehren der Mathematischen Wissenschaften, In Einzeldarstellungen mit Besonderer Berücksichtigung der Anwendungsgebiete |
| spelling | Radó, Tibor 1895-1965 Verfasser (DE-588)1025956125 aut Continuous Transformations in Analysis With an Introduction to Algebraic Topology by T. Rado, P. V. Reichelderfer Berlin, Heidelberg Springer Berlin Heidelberg 1955 1 Online-Ressource txt rdacontent c rdamedia cr rdacarrier Die Grundlehren der Mathematischen Wissenschaften, In Einzeldarstellungen mit Besonderer Berücksichtigung der Anwendungsgebiete 75 The general objective of this treatise is to give a systematic presentation of some of the topological and measure-theoretical foundations of the theory of real-valued functions of several real variables, with particular emphasis upon a line of thought initiated by BANACH, GEOCZE, LEBESGUE, TONELLI, and VITALI. To indicate a basic feature in this line of thought, let us consider a real-valued continuous function I(u) of the single real variable tt. Such a function may be thought of as defining a continuous translormation T under which x = 1 (u) is the image of u. About thirty years ago, BANACH and VITALI observed that the fundamental concepts of bounded variation, absolute continuity, and derivative admit of fruitful geometrical descriptions in terms of the transformation T: x = 1 (u) associated with the function 1 (u). They further noticed that these geometrical descriptions remain meaningful for a continuous transformation T in Euclidean n-space Rff, where T is given by a system of equations of the form 1-/(1 ff) X-I U, . . . ,tt ,. ", and n is an arbitrary positive integer. Accordingly, these geometrical descriptions can be used to define, for continuous transformations in Euclidean n-space Rff, n-dimensional concepts 01 bounded variation and absolute continuity, and to introduce a generalized Jacobian without reference to partial derivatives. These ideas were further developed, generalized, and modified by many mathematicians, and significant applications were made in Calculus of Variations and related fields along the lines initiated by GEOCZE, LEBESGUE, and TONELLI. Mathematics Mathematics, general Mathematik Analysis (DE-588)4001865-9 gnd rswk-swf Transformation Mathematik (DE-588)4060637-5 gnd rswk-swf Analysis (DE-588)4001865-9 s Transformation Mathematik (DE-588)4060637-5 s 1\p DE-604 Reichelderfer, Paul V. 1913-1996 Sonstige (DE-588)1116386216 oth Die Grundlehren der Mathematischen Wissenschaften, In Einzeldarstellungen mit Besonderer Berücksichtigung der Anwendungsgebiete 75 (DE-604)BV049758308 75 https://doi.org/10.1007/978-3-642-85989-2 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Radó, Tibor 1895-1965 Continuous Transformations in Analysis With an Introduction to Algebraic Topology Die Grundlehren der Mathematischen Wissenschaften, In Einzeldarstellungen mit Besonderer Berücksichtigung der Anwendungsgebiete Mathematics Mathematics, general Mathematik Analysis (DE-588)4001865-9 gnd Transformation Mathematik (DE-588)4060637-5 gnd |
| subject_GND | (DE-588)4001865-9 (DE-588)4060637-5 |
| title | Continuous Transformations in Analysis With an Introduction to Algebraic Topology |
| title_auth | Continuous Transformations in Analysis With an Introduction to Algebraic Topology |
| title_exact_search | Continuous Transformations in Analysis With an Introduction to Algebraic Topology |
| title_full | Continuous Transformations in Analysis With an Introduction to Algebraic Topology by T. Rado, P. V. Reichelderfer |
| title_fullStr | Continuous Transformations in Analysis With an Introduction to Algebraic Topology by T. Rado, P. V. Reichelderfer |
| title_full_unstemmed | Continuous Transformations in Analysis With an Introduction to Algebraic Topology by T. Rado, P. V. Reichelderfer |
| title_short | Continuous Transformations in Analysis |
| title_sort | continuous transformations in analysis with an introduction to algebraic topology |
| title_sub | With an Introduction to Algebraic Topology |
| topic | Mathematics Mathematics, general Mathematik Analysis (DE-588)4001865-9 gnd Transformation Mathematik (DE-588)4060637-5 gnd |
| topic_facet | Mathematics Mathematics, general Mathematik Analysis Transformation Mathematik |
| url | https://doi.org/10.1007/978-3-642-85989-2 |
| volume_link | (DE-604)BV049758308 |
| work_keys_str_mv | AT radotibor continuoustransformationsinanalysiswithanintroductiontoalgebraictopology AT reichelderferpaulv continuoustransformationsinanalysiswithanintroductiontoalgebraictopology |