Calculus of Several Variables (Undergraduate Texts in Mathematics)

✍ Scribed by Serge Lang

• Publisher: Springer
• Year: 2012
• Tongue: English
• Leaves: 628
• Edition: Softcover reprint of the original 3rd ed. 1987
• Category: Library

✦ Synopsis

This new, revised edition covers all of the basic topics in calculus of several variables, including vectors, curves, functions of several variables, gradient, tangent plane, maxima and minima, potential functions, curve integrals, Green’s theorem, multiple integrals, surface integrals, Stokes’ theorem, and the inverse mapping theorem and its consequences. It includes many completely worked-out problems.

✦ Table of Contents

Cover
Title page
Foreword
Contents
PART ONE Basic Material
CHAPTER I Vectors
�1. Definition of Points in Space
�2. Located Vectors
�3. Scalar Product
�4. The Norm of a Vector
�5. Parametric Lines
�6. Planes
�7. The Cross Product
CHAPTER II Differentiation of Vectors
�1. Derivative
�2. Length of Curves
CHAPTER III Functions of Several Variables
�1. Graphs and Level Curves
�2. Partial Derivatives
�3. Differentiability and Gradient
�4. Repeated Partial Derivatives
CHAPTER IV The Chain Rule and the Gradient
�1. The Chain Rule
�2. Tangent Plane
�3. Directional Derivative
�4. Functions Depending only on the Distance from the Origin
�5. The Law of Conservation of Energy
�6. Further Technique in Partial Differentiation
PART TWO Maxima, Minima, and Taylor's Formula
CHAPTER V Maximum and Minimum
�1. Critical Points
�2. Boundary Points
�3. Lagrange Multipliers
CHAPTER VI Higher Derivatives
�1. The First Two Terms in Taylor's Formula
�2. The Quadratic Term at Critical Points
�3. Algebraic Study of a Quadratic Form
�4. Partial Differential Operators
�5. The General Expression for Taylor's Formula
Appendix. Taylor's Formula in One Variable
Note. Chapter IX on Double Integrals is self contained, and could be covered here.
PART THREE Curve Integrals and Double Integrals
CHAPTER VII Potential Functions
�1. Existence and Uniqueness of Potential Functions
�2. Local Existence of Potential Functions
�3. An Important Special Vector Field
�4. Differentiating Under the Integral
�5. Proof of the Local Existence Theorem
CHAPTER VIII Curve Integrals
�1. Definition and Evaluation of Curve Integrals
�2. The Reverse Path
�3. Curve Integrals When the Vector Field Has a Potential Function
�4. Dependence of the Integral on the Path
CHAPTER IX Double Integrals
�1. Double Integrals
�2. Repeated Integrals
�3. Polar Coordinates
CHAPTER X Green's Theorem
�1. The Standard Version
�2. The Divergence and the Rotation of a Vector Field
PART FOUR Triple and Surface Integrals
CHAPTER XI Triple Integrals
�1. Triple Integrals
�2. Cylindrical and Spherical Coordinates
�3. Center of Mass
CHAPTER XII Surface Integrals
�1. Parametrization, Tangent Plane, and Normal Vector
�2. Surface Area
�3. Surface Integrals
�4. Curl and Divergence of a Vector Field
�5. Divergence Theorem in 3-Space
�6. Stokes' Theorem
PART FIVE Mappings, Inverse Mappings, and Change of Variables Formula
CHAPTER XIII Matrices
�1. Matrices
�2. Multiplication of Matrices
CHAPTER XIV Linear Mappings
�1. Mappings
�2. Linear Mappings
�3. Geometric Applications
�4. Composition and Inverse of Mappings
CHAPTER XV Determinants
�1. Determinants of Order 2
�2. Determinants of Order 3
�3. Additional Properties of Determinants
�4. Independence of Vectors
�5. Determinant of a Product
�6. Inverse of a Matrix
CHAPTER XVI Applications to Functions of Several Variables
�1. The Jacobian Matrix
�2. Differentiability
�3. The Chain Rule
�4. Inverse Mappings
�5. Implicit Functions
�6. The Hessian
CHAPTER XVII The Change of Variables Formula
�1. Determinants as Area and Volume
�2. Dilations
�3. Change of Variables Formula in Two Dimensions
�4. Application of Green's Formula to the Change of Variables Formula
�5. Change of Variables Formula in Three Dimensions
�6. Vector Fields on the Sphere
APPENDIX Fourier Series
�1. General Scalar Products
�2. Computation of Fourier Series
Answers
Index