Linear Methods of Applied Mathematics free book

Linear Methods of Applied Mathematics

Orthogonal series, boundary-value problems, and integral operators
Evans M. Harrell II and
James V. Herod

©
Copyright 1994,1995,1996, 1997, 2000 by Evans M. Harrell
II and James V. Herod. All rights reserved.

This is a WWW textbook written by Evans M. Harrell II and James V. Herod,
both of Georgia Tech. It is suitable for a first course on partial differential equations,
Fourier series and special functions, and integral equations. Students are expected
to have completed two years of calculus and an introduction to ordinary
differential equations and vector spaces. For recommended 10-week and
15-week syllabuses, read the
preface.

This text concentrates on mathematical concepts rather than on details of calculations,
which are often done with software, such as Maple or Mathematica.
It is not necessary to have experience
with
Maple or Mathematica
in order to read this text, nor is it the goal of this
text to teach software, but there are links in the text to
Maple worksheets and Mathematica notebooks, which perform calculations
and provide some supplementary instructive material. The supplementary
material exists both in a "flat" form, which can be read with Netscape, and
also in an active form, requiring mathematical software.

If you have access to mathematical software, you
may wish to take this opportunity to
set up the latest
version of Netscape to launch Mathematica or Maple automatically when appropriate.

You are welcome to browse, but if you make more than casual use, such as downloading
files or using them as study materials, certain restrictions and fees apply. Before
proceeding, please

Diagnostic quiz
Please take this before embarking on a course from this book.

Links to review materials on

ordinary differential equations and

linear algebra

Linearity

Also available in an
Adobe Acrobat version

The geometry of functions
Also available in an
Adobe Acrobat version

The red syllabus and the yellow syllabus continue with Chapter III
The green syllabus continues with

Chapter XIII
Fourier series. Introduction.
Also available in an
Adobe Acrobat version (without links)

Calculating Fourier series.
Also available in an
Adobe Acrobat version (without links)
test at this stage.

Differentiating Fourier series.
Also available in an
Adobe Acrobat version (without links)

The red syllabus continues with Chapter VI
The yellow syllabus continues with
Chapter XIII
Notes on a vibrating string.
Also available in an
Adobe Acrobat version (without links)

Traveling waves.
Also available in an
Adobe Acrobat version (without links)
test at this stage.

Mathematics of hot rods.
Also available in an
Adobe Acrobat version

PDEs in space. (includes potential equations)
Also available in an
Adobe Acrobat version (without links)

PDEs on a disk.
Also available in an
Adobe Acrobat version (without links)

test at this stage.
Great balls of PDEs.
Hunting for eigenvalues.

Geometry and integral operators.
Solving Y = KY + f.
test at this stage.

Ordinary differential operators.
Finding Green functions for ODEs.
test at this stage.

Green functions, Fourier series, and eigenfunctions.
Partial differential operators - classification and adjoints.
The free Green function and the method of images
test at this stage.
The fundamental solution of the heat equation
Using conformal mapping to construct Green functions.
Some advanced topics.

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Multivariable Calculus

This is a textbook for a course in multivariable calculus. It has been used for the past few years here at Georgia Tech. The notes are available as Adobe Acrobat documents,

this book also available online at http://www.math.gatech.edu/~cain/notes/calculus.html, written by George Cain and James Herod


Title page and Table of Contents

Table of Contents

Chapter One - Euclidean Three Space

1.1 Introduction

1.2 Coordinates in Three-Space

1.3 Some Geometry

1.4 Some More Geometry--Level Sets

Chapter Two - Vectors--Algebra and Geometry

  2.1 Vectors

  2.2 Scalar Product

  2.3 Vector Product

Chapter Three - Vector Functions

  3.1 Relations and Functions

  3.2 Vector Functions

  3.3 Limits and Continuity

Chapter Four - Derivatives

  4.1 Derivatives

  4.2 Geometry of Space Curves--Curvature

  4.3 Geometry of Space Curves--Torsion

  4.4 Motion

Chapter Five - More Dimensions

  5.1 The space Rⁿ

  5.2 Functions

Chapter Six - Linear Functions and Matrices

  6.1 Matrices

  6.2 Matrix Algebra

Chapter Seven - Continuity, Derivatives, and All That


  7.1 Limits and Continuity

  7.2 Derivatives

  7.3 The Chain Rule

Chapter Eight - f:Rⁿ-› R

  8.1 Introduction

  8.2 The Directional Derivative

  8.3 Surface Normals

  8.4 Maxima and Minima

  8.5 Least Squares

  8.6 More Maxima and Minima

  8.7 Even More Maxima and Minima

Chapter Nine - The Taylor Polynomial

  9.1 Introduction

  9.2 The Taylor Polynomial

  9.3 Error

  Supplementary material for Taylor polynomial in several variables.

Chapter Ten - Sequences, Series, and All That


  10.1 Introduction

  10.2 Sequences

  10.3 Series

  10.4 More Series

  10.5 Even More Series

  10.6 A Final Remark

Chapter Eleven - Taylor Series

  11.1 Power Series

  11.2 Limit of a Power Series

  11.3 Taylor Series

Chapter Twelve - Integration

  12.1 Introduction

  12.2 Two Dimensions

Chapter Thirteen - More Integration

  13.1 Some Applications

  13.2 Polar Coordinates

  13.3 Three Dimensions

Chapter Fourteen - One Dimension Again

  14.1 Scalar Line Integrals

  14.2 Vector Line Integrals

  14.3 Path Independence

Chapter Fifteen - Surfaces Revisited

  15.1 Vector Description of Surfaces

  15.2 Integration

Chapter Sixteen - Integrating Vector Functions

  16.1 Introduction

  16.2 Flux

Chapter Seventeen - Gauss and Green

  17.1 Gauss's Theorem

  17.2 Green's Theorem

  17.3 A Pleasing Application

Chapter Eighteen - Stokes

  18.1 Stokes's Theorem

  18.2 Path Independence Revisited

Chapter Ninteen - Some Physics

  19.1 Fluid Mechanics

  19.2 Electrostatics

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Calculus from Gilbert Strang

his book is legal to download and available online at http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm

TOCW is pleased to make this textbook available online. Published in 1991 and still in print from Wellesley-Cambridge Press, the book is a useful resource for educators and self-learners alike. It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications. There is also an online Instructor's Manual and a student Study Guide.

Textbook Components



Table of Contents (PDF)
Answers to Odd-Numbered Problems (PDF)
Equations (PDF)

ChapterS	FILES
1: Introduction to Calculus, pp. 1-43 1.1 Velocity and Distance, pp. 1-7 1.2 Calculus Without Limits, pp. 8-15 1.3 The Velocity at an Instant, pp. 16-21 1.4 Circular Motion, pp. 22-28 1.5 A Review of Trigonometry, pp. 29-33 1.6 A Thousand Points of Light, pp. 34-35 1.7 Computing in Calculus, pp. 36-43	Chapter 1 - complete (PDF - 4.1 MB) Chapter 1 - sections: 1.1 - 1.4 (PDF - 2.8 MB) 1.5 - 1.7 (PDF - 1.6 MB)
2: Derivatives, pp. 44-90 2.1 The Derivative of a Function, pp. 44-49 2.2 Powers and Polynomials, pp. 50-57 2.3 The Slope and the Tangent Line, pp. 58-63 2.4 Derivative of the Sine and Cosine, pp. 64-70 2.5 The Product and Quotient and Power Rules, pp. 71-77 2.6 Limits, pp. 78-84 2.7 Continuous Functions, pp. 85-90	Chapter 2 - complete (PDF - 4.3 MB) Chapter 2 - sections: 2.1 - 2.4 (PDF - 2.6 MB) 2.5 - 2.7 (PDF - 2.0 MB)
3: Applications of the Derivative, pp. 91-153 3.1 Linear Approximation, pp. 91-95 3.2 Maximum and Minimum Problems, pp. 96-104 3.3 Second Derivatives: Minimum vs. Maximum, pp. 105-111 3.4 Graphs, pp. 112-120 3.5 Ellipses, Parabolas, and Hyperbolas, pp. 121-129 3.6 Iterations x[n+1] = F(x[n]), pp. 130-136 3.7 Newton's Method and Chaos, pp. 137-145 3.8 The Mean Value Theorem and l'Hopital's Rule, pp. 146-153	Chapter 3 - complete (PDF - 5.9 MB) Chapter 3 - sections: 3.1 - 3.4 (PDF - 3.2 MB) 3.5 - 3.8 (PDF - 3.3 MB)
4: The Chain Rule, pp. 154-176 4.1 Derivatives by the Charin Rule, pp. 154-159 4.2 Implicit Differentiation and Related Rates, pp. 160-163 4.3 Inverse Functions and Their Derivatives, pp. 164-170 4.4 Inverses of Trigonometric Functions, pp. 171-176	Chapter 4 - complete (PDF - 2.0 MB) Chapter 4 - sections: 4.1 - 4.2 (PDF - 1.0 MB) 4.3 - 4.4 (PDF - 1.2 MB)
5: Integrals, pp. 177-227 5.1 The Idea of an Integral, pp. 177-181 5.2 Antiderivatives, pp. 182-186 5.3 Summation vs. Integration, pp. 187-194 5.4 Indefinite Integrals and Substitutions, pp. 195-200 5.5 The Definite Integral, pp. 201-205 5.6 Properties of the Integral and the Average Value, pp. 206-212 5.7 The Fundamental Theorem and Its Consequences, pp. 213-219 5.8 Numerical Integration, pp. 220-227	Chapter 5 - complete (PDF - 4.8 MB) Chapter 5 - sections: 5.1 - 5.4 (PDF - 2.2 MB) 5.5 - 5.8 (PDF - 2.8 MB)
6: Exponentials and Logarithms, pp. 228-282 6.1 An Overview, pp. 228-235 6.2 The Exponential e^x, pp. 236-241 6.3 Growth and Decay in Science and Economics, pp. 242-251 6.4 Logarithms, pp. 252-258 6.5 Separable Equations Including the Logistic Equation, pp. 259-266 6.6 Powers Instead of Exponentials, pp. 267-276 6.7 Hyperbolic Functions, pp. 277-282	Chapter 6 - complete (PDF - 4.9 MB) Chapter 6 - sections: 6.1 - 6.4 (PDF - 3.0 MB) 6.5 - 6.7 (PDF - 2.2 MB)
7: Techniques of Integration, pp. 283-310 7.1 Integration by Parts, pp. 283-287 7.2 Trigonometric Integrals, pp. 288-293 7.3 Trigonometric Substitutions, pp. 294-299 7.4 Partial Fractions, pp. 300-304 7.5 Improper Integrals, pp. 305-310	Chapter 7 - complete (PDF - 2.6 MB) Chapter 7 - sections: 7.1 - 7.3 (PDF - 1.7 MB) 7.4 - 7.5 (PDF - 1.0 MB)
8: Applications of the Integral, pp. 311-347 8.1 Areas and Volumes by Slices, pp. 311-319 8.2 Length of a Plane Curve, pp. 320-324 8.3 Area of a Surface of Revolution, pp. 325-327 8.4 Probability and Calculus, pp. 328-335 8.5 Masses and Moments, pp. 336-341 8.6 Force, Work, and Energy, pp. 342-347	Chapter 8 - complete (PDF - 3.4 MB) Chapter 8 - sections: 8.1 - 8.3 (PDF - 1.7 MB) 8.4 - 8.6 (PDF - 2.0 MB)
9: Polar Coordinates and Complex Numbers, pp. 348-367 9.1 Polar Coordinates, pp. 348-350 9.2 Polar Equations and Graphs, pp. 351-355 9.3 Slope, Length, and Area for Polar Curves, pp. 356-359 9.4 Complex Numbers, pp. 360-367	Chapter 9 - complete (PDF - 1.7 MB) Chapter 9 - sections: 9.1 - 9.2 (PDF) 9.3 - 9.4 (PDF - 1.0 MB)
10: Infinite Series, pp. 368-391 10.1 The Geometric Series, pp. 368-373 10.2 Convergence Tests: Positive Series, pp. 374-380 10.3 Convergence Tests: All Series, pp. 325-327 10.4 The Taylor Series for e^x, sin x, and cos x, pp. 385-390 10.5 Power Series, pp. 391-397	Chapter 10 - complete (PDF - 2.9 MB) Chapter 10 - sections: 10.1 - 10.3 (PDF - 1.9 MB) 10.4 - 10.5 (PDF - 1.2 MB)
11: Vectors and Matrices, pp. 398-445 11.1 Vectors and Dot Products, pp. 398-406 11.2 Planes and Projections, pp. 407-415 11.3 Cross Products and Determinants, pp. 416-424 11.4 Matrices and Linear Equations, pp. 425-434 11.5 Linear Algebra in Three Dimensions, pp. 435-445	Chapter 11 - complete (PDF - 4.0 MB) Chapter 11 - sections: 11.1 - 11.3 (PDF - 2.5 MB) 11.4 - 11.5 (PDF - 1.7 MB)
12: Motion along a Curve, pp. 446-471 12.1 The Position Vector, pp. 446-452 12.2 Plane Motion: Projectiles and Cycloids, pp. 453-458 12.3 Tangent Vector and Normal Vector, pp. 459-463 12.4 Polar Coordinates and Planetary Motion, pp. 464-471	Chapter 12 - complete (PDF - 2.2 MB) Chapter 12 - sections: 12.1 - 12.2 (PDF - 1.2 MB) 12.3 - 12.4 (PDF - 1.1 MB)
13: Partial Derivatives, pp. 472-520 13.1 Surface and Level Curves, pp. 472-474 13.2 Partial Derivatives, pp. 475-479 13.3 Tangent Planes and Linear Approximations, pp. 480-489 13.4 Directional Derivatives and Gradients, pp. 490-496 13.5 The Chain Rule, pp. 497-503 13.6 Maxima, Minima, and Saddle Points, pp. 504-513 13.7 Constraints and Lagrange Multipliers, pp. 514-520	Chapter 13 - complete (PDF - 4.9 MB) Chapter 13 - sections: 13.1 - 13.4 (PDF - 2.7 MB) 13.5 - 13.7 (PDF - 2.5 MB)
14: Multiple Integrals, pp. 521-548 14.1 Double Integrals, pp. 521-526 14.2 Changing to Better Coordinates, pp. 527-535 14.3 Triple Integrals, pp. 536-540 14.4 Cylindrical and Spherical Coordinates, pp. 541-548	Chapter 14 - complete (PDF - 2.5 MB) Chapter 14 - sections: 14.1 - 14.2 (PDF - 1.4 MB) 14.3 - 14.4 (PDF - 1.3 MB)
15: Vector Calculus, pp. 549-598 15.1 Vector Fields, pp. 549-554 15.2 Line Integrals, pp. 555-562 15.3 Green's Theorem, pp. 563-572 15.4 Surface Integrals, pp. 573-581 15.5 The Divergence Theorem, pp. 582-588 15.6 Stokes' Theorem and the Curl of F, pp. 589-598	Chapter 15 - complete (PDF - 4.3 MB) Chapter 15 - sections: 15.1 - 15.3 (PDF - 2.1 MB) 15.4 - 15.6 (PDF - 2.3 MB)
16: Mathematics after Calculus, pp. 599-615 16.1 Linear Algebra, pp. 599-602 16.2 Differential Equations, pp. 603-610 16.3 Discrete Mathematics, pp. 611-615	Chapter 16 - complete (PDF - 1.8 MB) Chapter 16 - sections: 16.1 - 16.2 (PDF - 1.5 MB) 16.3 (PDF)

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