nach oben

2014 | Buch

Kapitel lesen Erstes Kapitel lesen

Multivariate Calculus and Geometry

verfasst von: Seán Dineen

Verlag: Springer London

Buchreihe : Springer Undergraduate Mathematics Series

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

Multivariate calculus can be understood best by combining geometric insight, intuitive arguments, detailed explanations and mathematical reasoning. This textbook has successfully followed this programme. It additionally provides a solid description of the basic concepts, via familiar examples, which are then tested in technically demanding situations.

In this new edition the introductory chapter and two of the chapters on the geometry of surfaces have been revised. Some exercises have been replaced and others provided with expanded solutions.

Familiarity with partial derivatives and a course in linear algebra are essential prerequisites for readers of this book. Multivariate Calculus and Geometry is aimed primarily at higher level undergraduates in the mathematical sciences. The inclusion of many practical examples involving problems of several variables will appeal to mathematics, science and engineering students.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction to Differentiable Functions

Abstract

We introduce differentiable functions, directional and partial derivatives, graphs and level sets of functions of several variables.

Seán Dineen

Chapter 2. Level Sets and Tangent Spaces

Abstract

Using systems of linear equations as a guide we discuss the significance of the implicit function theorem for level sets. We define the tangent space and the normal space at a point on a level set.

Seán Dineen

Chapter 3. Lagrange Multipliers

Abstract

We develop the method of Lagrange multipliers to find the maximum and minimum of a function with constraints.

Seán Dineen

Chapter 4. Maxima and Minima on Open Sets

Abstract

We derive, using critical points and the Hessian, a method of locating local maxima, local minima and saddle points of a real-valued function defined on an open subset of ${\mathbb R}^n$.

Seán Dineen

Chapter 5. Curves in $${\mathbb {R}}^n$$

Abstract

We introduce and discuss the concept of directed curve in ${\mathbb R}^n$. We obtain a formula for the length of a curve, prove the existence of unit speed parametrizations and define piecewise smooth curves.

Seán Dineen

Chapter 6. Line Integrals

Abstract

We integrate vector-valued and scalar-valued functions along a directed curve in ${\mathbb R}^n$. We discuss scalar and vector potentials and define the curl of a vector field in ${\mathbb R}^3$.

Seán Dineen

Chapter 7. The Frenet–Serret Equations

Abstract

We discuss curvature and torsion of directed curves and derive the Frenet–Serret equations. Vector-valued differentiation and orthonormal bases are the main tools used.

Seán Dineen

Chapter 8. Geometry of Curves in $${\mathbb R}^3$$

Abstract

We apply the Frenet–Serret equations to study the geometric significance of torsion, to analyse curves in spheres and to characterise generalised helices.

Seán Dineen

Chapter 9. Double Integration

Abstract

We define the double integral of a function over an open subset of ${\mathbb R}^2$ and use Fubini’s theorem to evaluate such integrals. We discuss the fundamental theorem of calculus in ${\mathbb R}^2$—Green’s theorem.

Seán Dineen

Chapter 10. Parametrized Surfaces in $${\mathbb R}^3$$

Abstract

We discuss theoretical and practical approaches to parametrizing a surface in ${\mathbb R}^3$.

Seán Dineen

Chapter 11. Surface Area

Abstract

We define and calculate surface area.

Seán Dineen

Chapter 12. Surface Integrals

Abstract

We define the integral of a vector field over an oriented surface. Geometrical interpretations are discussed.

Seán Dineen

Chapter 13. Stokes’ Theorem

Abstract

We discuss Stokes’ theorem for oriented surfaces in ${\mathbb R}^3$.

Seán Dineen

Chapter 14. Triple Integrals

Abstract

We define triple integrals of scalar-valued functions over open subsets of ${\mathbb R}^3$, discuss coordinate systems in ${\mathbb R}^3$, justify a change of variable formula and use Fubini’s theorem to evaluate integrals.

Seán Dineen

Chapter 15. The Divergence Theorem

Abstract

We state, discuss and give examples of the divergence theorem of Gauss.

Seán Dineen

Chapter 16. Geometry of Surfaces in $${\mathbb {R}}^3$$

Abstract

Using normal sections we define normal curvature, principal curvatures and Gaussian curvature. Geometric interpretations and a method of calculating the Gaussian curvature using parametrization are given.

Seán Dineen

Chapter 17. Gaussian Curvature

Abstract

We define the Weingarten mapping or shape operator. This gives an intrinsic approach to Gaussian curvature.

Seán Dineen

Chapter 18. Geodesic Curvature

Abstract

We define geodesic curvature and geodesics. For a curve on a surface we derive a formula connecting intrinsic curvature, normal curvature and geodesic curvature. We discuss paths of shortest distance, further interpretations of Gaussian curvature and introduce, informally and geometrically, a number of important results in differential geometry.

Seán Dineen

Backmatter

Titel: Multivariate Calculus and Geometry
verfasst von: Seán Dineen
Verlag: Springer London
Electronic ISBN: 978-1-4471-6419-7
Print ISBN: 978-1-4471-6418-0
DOI: https://doi.org/10.1007/978-1-4471-6419-7