Buch, Englisch, 319 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 1030 g
ISBN: 978-0-8176-4274-7
Verlag: Birkhäuser Boston
This work provides a systematic examination of derivatives and integrals of multivariable functions. The approach taken here is similar to that of the author’s previous "Continuous Functions of Vector Variables": specifically, elementary results from single-variable calculus are extended to functions in several-variable Euclidean space. Topics encompass differentiability, partial derivatives, directional derivatives and the gradient; curves, surfaces, and vector fields; the inverse and implicit function theorems; integrability and properties of integrals; and the theorems of Fubini, Stokes, and Gauss. Prerequisites include background in linear algebra, one-variable calculus, and some acquaintance with continuous functions and the topology of the real line.
Written in a definition-theorem-proof format, the book is replete with historical comments, questions, and discussions about strategy, difficulties, and alternate paths. "Derivatives and Integrals of Multivariable Functions" is a rigorous introduction to multivariable calculus that will help students build a foundation for further explorations in analysis and differential geometry.
Zielgruppe
Graduate
Autoren/Hrsg.
Fachgebiete
- Mathematik | Informatik Mathematik Mathematische Analysis Harmonische Analysis, Fourier-Mathematik
- Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
- Mathematik | Informatik Mathematik Mathematische Analysis Elementare Analysis und Allgemeine Begriffe
- Mathematik | Informatik Mathematik Mathematische Analysis Reelle Analysis
- Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis
Weitere Infos & Material
1 Differentiability of Multivariate Functions.- 1.1 Differentiability.- 1.2 Derivatives and Partial Derivatives.- 1.3 The Chain Rule.- 1.4 Higher Derivatives.- 2 Derivatives of Scalar Functions.- 2.1 Directional Derivatives and the Gradient.- 2.2 The Mean Value Theorem.- 2.3 Extreme Values and the Derivative.- 2.4 Extreme Values and the Second Derivative.- 2.5 Implicit Scalar Functions.- 2.6 Curves, Surfaces, Tangents, and Normals.- 3 Derivatives of Vector Functions.- 3.1 Contractions.- 3.2 The Inverse Function Theorem.- 3.3 The Implicit Function Theorem.- 3.4 Lagrange’s Method.- 4 Integrability of Multivariate Functions.- 4.1 Partitions.- 4.2 Integrability in a Box.- 4.3 Domains of Integrability.- 4.4 Integrability and Sets of Zero Volume.- 5 Integrals of Scalar Functions.- 5.1 Fubini's Theorem.- 5.2 Properties of Integrals.- 5.3 Change of Variable.- 5.4 Generalized Integrals.- 5.5 Line Integrals.- 5.6 Surface Integrals.- 6 Vector Integrals and the Vector-Field Theorems.- 6.1 Integrals of the Tangential and Normal Components.- 6.2 Path-Independence.- 6.3 On the Edge: The Theorems of Green and Stokes.- 6.4 Gauss's Theorem.- Solutions to Exercises.- References.
