Multivariable Calculus: A Linear Algebra Based Approach
Edition: 1
Copyright: 2021
Pages: 310
Edition: 1
Copyright: 2021
Pages: 310 / LSI pg cnt does not change
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Multivariable Calculus: A Linear Algebra Based Approach provides a unique introduction to the standard topics of vector calculus for students who have completed an introductory linear algebra course. Topics covered include vector-valued functions of a real variable, real-valued functions of a vector, vector differential calculus, multiple integrals, calculus along curves, and vector field theory through the theorems of Gauss, Green and Stokes, concluding with an introduction to differential forms. Linear algebra provides an appropriate perspective on this material. We stress, for examples, how the derivative is really a matrix, the chain rule amounts to matrix multiplication, and positive definite matrices explain the second derivative test.
Preface
1 Remembrance of Things Past
1.1 Calculus
1.1.1 Limits, Continuity, and the Derivative
1.1.2 Properties and Uses of the Derivative
1.1.3 The Definite Integral
1.1.4 Fundamental Theorem of Calculus
1.1.5 Taylor’s Theorem
1.2 Linear Algebra
1.2.1 Vectors
1.2.2 Linear Independence
1.2.3 Matrices
1.2.4 Eigenvalues and Eigenvectors
1.3 Exercises and Projects
2 Vector-Valued Functions of One Variable
2.1 Curves in the Plane and Space
2.2 Limits and Continuity
2.3 Derivatives
2.4 Velocity, Speed, and Acceleration
2.5 Integrals
2.6 Applications
2.6.1 Projectile Motion
2.6.2 Kepler’s Laws of Planetary Motion
2.7 Exercises and Projects
3 Real-Valued Functions of a Vector: The Derivative
3.1 Some Examples
3.2 Graphs and Level Sets
3.3 Partial Derivatives
3.4 Parametrized Surfaces
3.5 Applications
3.5.1 Utility Functions
3.5.2 An Age-Structured Population Model
3.6 Exercises and Projects
4 Differentiable Functions
4.1 Limits and Continuity
4.2 Differentiability
4.3 Directional Derivatives
4.4 A Mean Value Theorem
4.5 The Jacobian Matrix
4.6 Applications
4.6.1 Economic Growth
4.6.2 Newton’s Method
4.7 Exercises and Projects
5 Vector Differential Calculus
5.1 Differentiating Compositions of Functions
5.1.1 The Little Chain Rule
5.1.2 General Chain Rule
5.2 Change of Variables
5.3 Gradient Fields
5.4 Normal Vectors
5.5 Implicit Differentiation
5.6 Extreme Values
5.6.1 Critical Values
5.6.2 Method of Lagrange Multipliers
5.6.3 Second Derivative Criteria
5.7 Alternative Coordinate Systems
5.7.1 Polar Coordinates
5.7.2 Cylindrical Coordinates
5.7.3 Spherical Coordinates
5.8 Applications
5.8.1 Maximizing Utility With Budget Constraint
5.8.2 Laplace Equation
5.9 Exercises and Projects
6 Multiple Integrals
6.1 The Iterated Integral
6.2 The Multiple Integral
6.2.1 Definition
6.2.2 Existence
6.2.3 Double Integrals
6.2.4 Triple Integrals
6.3 Properties of the Integral
6.4 Jacobians and the Change of Variable
6.5 Improper Integrals
6.6 Applications
6.6.1 Probability
6.6.2 Density and Moments
6.7 Exercises and Projects
7 Calculus Along Curves
7.1 Work
7.2 Vector Fields and Line Integrals
7.2.1 Line Integrals
7.3 Arc Length and Weighted Curves
7.3.1 Weighted Curves
7.4 Surfaces of Revolution
7.5 Numerical Integration
7.6 Curvature and Normals
7.6.1 Curvature
7.7 Flow Lines and Differential Equations
7.7.1 Alternative Notation for Vector Fields
7.8 Applications
7.9 Exercises and Projects
8 Vector Field Theory
8.1 Divergence and Curl
8.1.1 Divergence of a Vector Field
8.1.2 Properties of Divergence
8.1.3 The Del Operator
8.1.4 Curl
8.1.5 Derivative Identities
8.2 Conservative Vector Fields
8.2.1 Potential Functions
8.2.2 Equivalent Notions of Path Equivalence
8.2.3 Symmetric Jacobians and Path Independence
8.3 Green’s Theorem in the Plane
8.3.1 Setting for Green’s Theorem
8.3.2 A Verification of Green’s Theorem
8.3.3 Proof of Green’s Theorem for a Very Simple Region
8.3.4 Green’s Theorem for Simple Regions
8.3.5 Green’s Theorem for More Complicated Regions
8.3.6 Using Green’s Theorem to Evaluate Line Integrals
8.3.7 The Divergence Theorem
8.3.8 Jacobian Symmetry and Gradient Fields
8.3.9 Finding Potentials Using Partial Integration
8.4 Surface Integrals
8.4.1 Mass
8.4.2 Integrating Vector Fields Over Surfaces
8.4.3 Orientation
8.5 Gauss’s Theorem
8.5.1 Gauss’s Theorem for Simple Regions
8.5.2 Surface Independence
8.5.3 Meaning of Divergence
8.6 Stokes’ Theorem
8.6.1 Proving Stokes’ Theorem
8.6.2 Simply Connected Surfaces and Conservative Fields
8.6.3 Interpreting Curl
8.6.4 Independence of Surface
8.6.5 An Historical Note
8.7 Applications
8.7.1 Geometry: Shoelace Theorem
8.7.2 Newton’s Law of Gravitational Attraction
8.8 Exercises and Projects
9 Differential Forms and Vector Calculus
9.1 What Are Differential Forms?
9.1.1 0-Forms
9.1.2 1-Forms
9.1.3 Integrating a 1-Form Over a Curve
9.1.4 2-Forms
9.1.5 Integrating a 2-Form Over a Surface
9.1.6 3-forms
9.1.7 Integrating a 3-Form Over a Region
9.2 Algebra of Forms
9.3 Differentiating Forms
9.4 Generalizing the Fundamental Theorem of Calculus
9.4.1 Gauss’ Theorem
9.4.2 Stokes’ Theorem
9.5 Exercises and Projects
Index
Michael Olinick earned a BA in mathematics and philosophy from the University of Michigan and completed his MA and PhD at the University of Wisconsin in Madison. He teaches in the Mathematics and Statistics department at Middlebury College where he has served as an Alfred P. Sloan Resource Professor and Baldwin Professor of Mathematics and Natural Philosophy. He has also authored or co-authored books on mathematical modeling, calculus of one and several variables, probabilistic reasoning, principles and practice of mathematics, and the life and work of Alan Turing. Olinick has held visiting appointments at the University of East Africa’s University College Nairobi, University of California Berkeley, Wesleyan University, and Lancaster University.