Differential and integral calculus / by R. Courant ; translated by E. J. McShane.
Idioma: Inglés Lenguaje original: Alemán Editor: London : Blackie & Son, 1934-1936Descripción: 2 v. : il. ; 23 cmTítulos uniformes: Vorlesungen über Differential- und Integralrechnung. Inglés Otra clasificación: 26-01CONTENTS Chapter I PRELIMINARY REMARKS ON ANALYTICAL GEOMETRY AND VECTOR ANALYSIS 1. Rectangular Co-ordinates and Vectors [1] 2. The Area of a Triangle, the Volume of a Tetrahedron, the Vector Multiplication of Vectors [12] 3. Simple Theorems on Determinants of the Second and Third Order [19] 4. Affine Transformations and the Multiplication of Determinants [27] Chapter II FUNCTIONS OF SEVERAL VARIABLES AND THEIR DERIVATIVES 1. The Concept of Function in the Case of Several Variables [39] 2. Continuity [44] 3. The Derivatives of a Function [50] 4. The Total Differential of a Function and its Geometrical Meaning [59] 5. Functions of Functions (Compound Functions) and the Introduction of New Independent Variables [69] 6. The Mean Value Theorem and Taylor’s Theorem for Functions of Several Variables [78] 7. The Application of Vector Methods [82] APPENDIX 1. The Principle of the Point of Accumulation in Several Dimensions and its Applications [95] 2. The Concept of Limit for Functions of Several Variables [101] 3. Homogeneous Functions [108] Chapter III DEVELOPMENTS AND APPLICATIONS OF THE DIFFERENTIAL CALCULUS 1. Implicit Functions [111] 2. Curves and Surfaces in Implicit Form [122] 3. Systems of Functions, Transformations, and Mappings [133] 4. Applications [159] 5. Families of Curves, Families of Surfaces, and their Envelopes [169] 6. Maxima and Minima [183] APPENDIX 1. Sufficient Conditions for Extreme Values [204] 2. Singular Points of Plane Curves [209] 3. Singular Points of Surfaces [211] 4. Connexion between Euler’s and Lagrange’s Representations of the Motion of a Fluid [212] 5. Tangential Representation of a Closed Curve [213] Chapter IV MULTIPLE INTEGRALS 1. Ordinary Integrals as Functions of a Parameter [215] 2. The Integral of a Continuous Function over a Region of the Plane or of Space [223] 3. Reduction of the Multiple Integral to Repeated Single Integrals [236] 4. Transformation of Multiple Integrals [247] 5. Improper Integrals [256] 6. Geometrical Applications [264] 7. Physical Applications [276] APPENDIX 1. The Existence of the Multiple Integral [287] 2. General Formula for the Area (or Volume) of a Region bounded by Segments of Straight Lines or Plane Areas (Guldin’s Formula). The Polar Planimeter [294] 3. Volumes and Areas in Space of any Number of Dimensions [298] 4. Improper Integrals as Functions of a Parameter [307] 5. The Fourier Integral [318] 6. The Eulerian Integrals (Gamma Function) [323] 7. Differentiation and Integration to Fractional Order. Abel’s Integral Equation [339] 8. Note on the Definition of the Area of a Curved Surface [341] Chapter V INTEGRATION OVER REGIONS IN SEVERAL DIMENSIONS 1. Line Integrals [343] 2. Connexion between Line Integrals and Double Integrals in the Plane. (The Integral Theorems of Gauss, Stokes, and Green) [359] 3. Interpretation and Applications of the Integral Theorems for the Plane [370] 4. Surface Integrals [374] 5. Gauss’s Theorem and Green’s Theorem in Space [384] 6. Stokes’s Theorem in Space [392] 7. The Connexion between Differentiation and Integration for Several Variables [397] APPENDIX 1. Remarks on Gauss’s Theorem and Stokes’s Theorem [402] 2. Representation of a Source-free Vector Field as a Curl [404] Chapter VI DIFFERENTIAL EQUATIONS 1. The Differential Equations of the Motion of a Particle in Three Dimensions [412] 2. Examples on the Mechanics of a Particle [418] 3. Further Examples of Differential Equations [429] 4. Linear Differential Equations [438] 5. General Remarks on Differential Equations [450] 6. The Potential of Attracting Charges [468] 7. Further Examples of Partial Differential Equations [481] Chapter VII CALCULUS OF VARIATIONS 1. Introduction [491] 2. Euler’s Differential Equation in the Simplest Case [497] 3. Generalizations [507] Chapter VIII FUNCTIONS OF A COMPLEX VARIABLE 1. Introduction [522] 2. Foundations of the Theory of Functions of a Complex Variable [530] 3. The Integration of Analytic Functions [537] 4. Cauchy’s Formula and its Applications [545] 5. Applications to Complex Integration (Contour Integration) [554] 6. Many-valued Functions and Analytic Extension [563] SUPPLEMENT Real Numbers and the Concept of Limit [569] Miscellaneous Examples [587] Summary of Important Theorems and Formulae [600] Answers and Hints [623] Index [679]
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 26 C858 (Browse shelf) | Vol. II | Available | A-2343 |
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| 26 C836a Análise matemática | 26 C836a Análise matemática | 26 C836a Análise matemática | 26 C858 Differential and integral calculus / | 26 C858-2 Differential and integral calculus. | 26 C858i Introduction to calculus and analysis / | 26 C858i Introduction to calculus and analysis / |
Traducción de: Vorlesungen über Differential- und Integralrechnung.
La biblioteca sólo posee el vol. II de esta edición. Para el vol. I, véase la 2da edición. AR-BbIMB
CONTENTS --
Chapter I --
PRELIMINARY REMARKS ON ANALYTICAL GEOMETRY AND VECTOR ANALYSIS --
1. Rectangular Co-ordinates and Vectors [1] --
2. The Area of a Triangle, the Volume of a Tetrahedron, the Vector Multiplication of Vectors [12] --
3. Simple Theorems on Determinants of the Second and Third Order [19] --
4. Affine Transformations and the Multiplication of Determinants [27] --
Chapter II --
FUNCTIONS OF SEVERAL VARIABLES AND THEIR DERIVATIVES --
1. The Concept of Function in the Case of Several Variables [39] --
2. Continuity [44] --
3. The Derivatives of a Function [50] --
4. The Total Differential of a Function and its Geometrical Meaning [59] --
5. Functions of Functions (Compound Functions) and the Introduction of New Independent Variables [69] --
6. The Mean Value Theorem and Taylor’s Theorem for Functions of Several Variables [78] --
7. The Application of Vector Methods [82] --
APPENDIX --
1. The Principle of the Point of Accumulation in Several Dimensions and its Applications [95] --
2. The Concept of Limit for Functions of Several Variables [101] --
3. Homogeneous Functions [108] --
Chapter III --
DEVELOPMENTS AND APPLICATIONS OF THE DIFFERENTIAL CALCULUS --
1. Implicit Functions [111] --
2. Curves and Surfaces in Implicit Form [122] --
3. Systems of Functions, Transformations, and Mappings [133] --
4. Applications [159] --
5. Families of Curves, Families of Surfaces, and their Envelopes [169] --
6. Maxima and Minima [183] --
APPENDIX --
1. Sufficient Conditions for Extreme Values [204] --
2. Singular Points of Plane Curves [209] --
3. Singular Points of Surfaces [211] --
4. Connexion between Euler’s and Lagrange’s Representations of the Motion of a Fluid [212] --
5. Tangential Representation of a Closed Curve [213] --
Chapter IV --
MULTIPLE INTEGRALS --
1. Ordinary Integrals as Functions of a Parameter [215] --
2. The Integral of a Continuous Function over a Region of the Plane or of Space [223] --
3. Reduction of the Multiple Integral to Repeated Single Integrals [236] --
4. Transformation of Multiple Integrals [247] --
5. Improper Integrals [256] --
6. Geometrical Applications [264] --
7. Physical Applications [276] --
APPENDIX --
1. The Existence of the Multiple Integral [287] --
2. General Formula for the Area (or Volume) of a Region bounded by Segments of Straight Lines or Plane Areas (Guldin’s Formula). The Polar Planimeter [294] --
3. Volumes and Areas in Space of any Number of Dimensions [298] --
4. Improper Integrals as Functions of a Parameter [307] --
5. The Fourier Integral [318] --
6. The Eulerian Integrals (Gamma Function) [323] --
7. Differentiation and Integration to Fractional Order. Abel’s Integral Equation [339] --
8. Note on the Definition of the Area of a Curved Surface [341] --
Chapter V --
INTEGRATION OVER REGIONS IN SEVERAL DIMENSIONS --
1. Line Integrals [343] --
2. Connexion between Line Integrals and Double Integrals in the Plane. (The Integral Theorems of Gauss, Stokes, and Green) [359] --
3. Interpretation and Applications of the Integral Theorems for the Plane [370] --
4. Surface Integrals [374] --
5. Gauss’s Theorem and Green’s Theorem in Space [384] --
6. Stokes’s Theorem in Space [392] --
7. The Connexion between Differentiation and Integration for Several Variables [397] --
APPENDIX --
1. Remarks on Gauss’s Theorem and Stokes’s Theorem [402] --
2. Representation of a Source-free Vector Field as a Curl [404] --
Chapter VI --
DIFFERENTIAL EQUATIONS --
1. The Differential Equations of the Motion of a Particle in Three Dimensions [412] --
2. Examples on the Mechanics of a Particle [418] --
3. Further Examples of Differential Equations [429] --
4. Linear Differential Equations [438] --
5. General Remarks on Differential Equations [450] --
6. The Potential of Attracting Charges [468] --
7. Further Examples of Partial Differential Equations [481] --
Chapter VII --
CALCULUS OF VARIATIONS --
1. Introduction [491] --
2. Euler’s Differential Equation in the Simplest Case [497] --
3. Generalizations [507] --
Chapter VIII --
FUNCTIONS OF A COMPLEX VARIABLE --
1. Introduction [522] --
2. Foundations of the Theory of Functions of a Complex Variable [530] --
3. The Integration of Analytic Functions [537] --
4. Cauchy’s Formula and its Applications [545] --
5. Applications to Complex Integration (Contour Integration) [554] --
6. Many-valued Functions and Analytic Extension [563] --
SUPPLEMENT --
Real Numbers and the Concept of Limit [569] --
Miscellaneous Examples [587] --
Summary of Important Theorems and Formulae [600] --
Answers and Hints [623] --
Index [679] --
MR, REVIEW #
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