B. Craven

Functions of several variables (eBook, PDF)

• Format: PDF

Produktbeschreibung

This book is aimed at mathematics students, typically in the second year of a university course. The first chapter, however, is suitable for first-year students. Differentiable functions are treated initially from the standpoint of approximating a curved surface locally by a fiat surface. This enables both geometric intuition, and some elementary matrix algebra, to be put to effective use. In Chapter 2, the required theorems - chain rule, inverse and implicit function theorems, etc- are stated, and proved (for n variables), concisely and rigorously. Chapter 3 deals with maxima and minima, including problems with equality and inequality constraints. The chapter includes criteria for discriminating between maxima, minima and saddlepoints for constrained problems; this material is relevant for applications, but most textbooks omit it. In Chapter 4, integration over areas, volumes, curves and surfaces is developed, and both the change-of-variable formula, and the Gauss-Green-Stokes set of theorems are obtained. The integrals are defined with approximative sums (ex pressed concisely by using step-functions); this preserves some geometrical (and physical) concept of what is happening. Consequent on this, the main ideas of the 'differential form' approach are presented, in a simple form which avoids much of the usual length and complexity. Many examples and exercises are included.

---

Dieser Download kann aus rechtlichen Gründen nur mit Rechnungsadresse in A, B, BG, CY, CZ, D, DK, EW, E, FIN, F, GR, HR, H, IRL, I, LT, L, LR, M, NL, PL, P, R, S, SLO, SK ausgeliefert werden.

Produktdetails

• Verlag: Springer Netherlands
• Seitenzahl: 138
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9789401093477
• Artikelnr.: 44064115

Inhaltsangabe

1. Differentiable Functions.- 1.1 Introduction.- 1.2 Linear part of a function.- 1.3 Vector viewpoint.- 1.4 Directional derivative.- 1.5 Tangent plane to a surface.- 1.6 Vector functions.- 1.7 Functions of functions.- 2. Chain Rule and Inverse Function Theorem.- 2.1 Norms.- 2.2 Fréchet derivatives.- 2.3 Chain rule.- 2.4 Inverse function theorem.- 2.5 Implicit functions.- 2.6 Functional dependence.- 2.7 Higher derivatives.- 3. Maxima and Minima.- 3.1 Extrema and stationary points.- 3.2 Constrained minima and Lagrange multipliers.- 3.3 Discriminating constrained stationary points.- 3.4 Inequality constraints.- 3.5 Discriminating maxima and minima with inequality constraints 62 Further reading.- 4. Integrating Functions of Several Variables.- 4.1 Basic ideas of integration.- 4.2 Double integrals.- 4.3 Length, area and volume.- 4.4 Integrals over curves and surfaces.- 4.5 Differential forms.- 4.6 Stokes's theorem.- Further reading.- Appendices.- A. Background required in linear algebra and elementary calculus.- B. Compact sets, continuous functions and partitions of unity.- C. Answers to selected exercises.- Index (including table of some special symbols).