Alignment: Calculus • Volume 2 • Multi Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability • Second Edition

Table of Contents

  • Part 1 Linear Analysis
    • 1 Linear Spaces
      • 1.1 Introduction
      • 1.2 The Definition of a Linear Space
      • 1.3 Examples of Linear Spaces
      • 1.4 Elementary Consequences of the Axioms
      • 1.5 Exercises
      • 1.6 Subspaces of a Linear Space
      • 1.7 Dependent and Independent Sets in a Linear Space
      • 1.8 Bases and Dimension
      • 1.9 Components
      • 1.10 Exercises
      • 1.11 Inner Products, Euclidean Spaces. Norms
      • 1.12 Orthogonality in a Euclidean Space
      • 1.13 Exercises
      • 1.14 Construction of Orthogonal Sets. The Gram-Schmidt Process
      • 1.15 Orthogonal Complements. Projections
      • 1.16 Best Approximation of Elements in a Euclidean Space by Elements in a Finite-Dimensional Subspace
    • 2 Linear Transformations and Matrices
      • 2.1 Linear Transformations
      • 2.2 Null Space and Range
      • 2.3 Nullity and Rank
      • 2.4 Exercises
      • 2.5 Algebraic Operations on Linear Transformations
      • 2.6 Inverses
      • 2.7 One-to-One Linear Transformations
      • 2.8 Exercises
      • 2.9 Linear Transformations with Prescribed Values
      • 2.10 Matrix Representations of Linear Transformations
      • 2.11 Construction of a Matrix Representation in Diagonal Form
      • 2.12 Exercises
      • 2.13 Linear Spaces of Matrices
      • 2.14 Tsomorphism between Linear Transformations and Matrices
      • 2.15 Multiplication of Matrices
      • 2.16 Exercises
      • 2.17 Systems of Linear Equations
      • 2.18 Computation Techniques
      • 2.19 Inverses of Square Matrices
      • 2.20 Exercises
      • 2.21 Miscellaneous Exercises on Matrices
    • 3 Determinants
      • 3.1 Introduction
      • 3.2 Motivation for the Choice of Axioms for a Determinant Function
      • 3.3 A Set of Axioms for a Determinant Function
      • 3.4 Computation of Determinants
      • 3.5 The Uniqueness Theorem
      • 3.6 Exercises
      • 3.7 The Product Formula for Determinants
      • 3.8 The Determinant of the Inverse of a Nonsingular Matrix
      • 3.9 Determinants and Independence of Vectors
      • 3.10 The Determinant of a Block-Diagonal Matrix
      • 3.11 Exercises
      • 3.12 Expansion Formulas for Determinants. Minors and Cofactors
      • 3.13 Existence of the Determinant Function
      • 3.14 The Determinant of a Transpose
      • 3.15 The Cofactor Matrix
      • 3.16 Cramer’s Rule
    • 4 Eigenvalues and Eigenvectors
      • 4.1 Linear Transformations with Diagonal Matrix Representations
      • 4.2 Eigenvectors and Eigenvalues of a Linear Transformation
      • 4.3 Linear Independence of Eigenvectors Corresponding to Distinct Eigenvalues
      • 4.4 Exercises
      • 4.5 The Finite-Dimensional Case. Characteristic Polynomials
      • 4.6 Calculation of Eigenvalues and Eigenvectors in the Finite-Dimensional Case
      • 4.7 Trace of a Matrix
      • 4.8 Exercises
      • 4.9 Matrices Representing the Same Linear Transformation. Similar Matrices
    • 5 Eigenvalues of Operators Acting on Euclidean Spaces
      • 5.1 Eigenvalues and Inner Products
      • 5.2 Hermitian and Skew-Hermitian Transformations
      • 5.3 Eigenvalues and Eigenvectors of Hermitian and Skew-Hermitian Operators
      • 5.4 Orthogonality of Eigenvectors Corresponding to Distinct Eigenvalues
      • 5.5 Exercises
      • 5.6 Existence of an Orthonormal Set of Eigenvectors for Hermitian and Skew-Hermitian Operators Acting on Finite-Dimensional Spaces
      • 5.7 Matrix Representations for Hermitian and Skew-Hermitian Operators
      • 5.8 Hermitian and Skew-Hermitian Matrices. The Adjoint of a Matrix
      • 5.9 Diagonalization of a Hermitian or Skew-Hermitian Matrix
      • 5.10 Unitary Matrices. Orthogonal Matrices
      • 5.11 Exercises
      • 5.12 Quadratic Forms
      • 5.13 Reduction of a Real Quadratic Form to a Diagonal Form
      • 5.14 Applications to Analytic Geometry
      • 5.15 Exercises
      • 5.16 Eigenvalues of a Symmetric Transformation Obtained as Values of Its Quadratic Form
      • 5.17 Extremal Properties of Eigenvalues of a Symmetric Transformation
      • 5.18 The Finite-Dimensional Case
      • 5.19 Unitary Transformations
    • 6 Linear Differential Equations
      • 6.1 Historical Introduction
      • 6.2 Review of Results Concerning Linear Equations of First and Second Orders
      • 6.3 Exercises
      • 6.4 Linear Differential Equations of Order 𝑛
      • 6.5 The Existence-Uniqueness Theorem
      • 6.6 The Dimension of the Solution Space of a Homogeneous Linear Equation
      • 6.7 The Algebra of Constant-Coefficient Operators
      • 6.8 Determination of a Basis of Solutions for Linear Equations with Constant Coefficients by Factorization of Operators
      • 6.9 Exercises
      • 6.10 The Relation between the Homogeneous and Nonhomogeneous Equations
      • 6.11 Determination of a Particular Solution of the Nonhomogeneous Equation. The Method of Variation of Parameters
      • 6.12 Nonsingularity of the Wronskian Matrix of 𝑛 Independent Solutions of a Homogeneous Linear Equation
      • 6.13 Special Methods for Determining a Particular Solution of the Nonhomogeneous Equation. Reduction to a System of First-Order Linear Equations
      • 6.14 The Annihilator Method for Determining a Particular Solution of the Nonhomogeneous Equation
      • 6.15 Exercises
      • 6.16 Miscellaneous Exercises on Linear Differential Equations
      • 6.17 Linear Equations of Second Order with Analytic Coefficients
      • 6.18 The Legendre Equation
      • 6.19 The Legendre Polynomials
      • 6.20 Rodrigues’ Formula for the Legendre Polynomials
      • 6.21 Exercises
      • 6.22 The Method of Frobenius
      • 6.23 The Bessel Equation
    • 7 Systems of Differential Equations
      • 7.1 Introduction
      • 7.2 Calculus of Matrix Functions
      • 7.3 Infinite Series of Matrices. Norms of Matrices
      • 7.4 Exercises
      • 7.5 The Exponential Matrix
      • 7.6 The Differential Equation Satisfied by 𝑒^𝑡𝐴
      • 7.7 Uniqueness Theorem for the Matrix Differential Equation 𝐹′(𝑡) = 𝐴𝐹(𝑡)
      • 7.8 The Law of Exponents for Exponential Matrices
      • 7.9 Existence and Uniqueness Theorems for Homogeneous Linear Systems with Constant Coefficients
      • 7.10 The Problem of Calculating 𝑒^𝑡𝐴
      • 7.11 The Cayley-Hamilton Theorem
      • 7.12 Exercises
      • 7.13 Putzer’s Method for Calculating 𝑒^𝑡𝐴
      • 7.14 Alternate Methods for Calculating 𝑒^𝑡𝐴 in Special Cases
      • 7.15 Exercises
      • 7.16 Nonhomogeneous Linear Systems with Constant Coefficients
      • 7.17 Exercises
      • 7.18 The General Linear System 𝑌′(𝑡) = 𝑃(𝑡)𝑌(𝑡) + 𝑄(𝑡)
      • 7.19 A Power-Series Method for Solving Homogeneous Linear Systems
      • 7.20 Exercises
      • 7.21 Proof of the Existence Theorem by the Method of Successive Approximations
      • 7.22 The Method of Successive Approximations Applied to First-Order Nonlinear Systems
      • 7.23 Proof of an Existence-Uniqueness Theorem for First-Order Nonlinear Systems
      • 7.24 Exercises
      • 7.25 Successive Approximations and Fixed Points of Operators
      • 7.26 Normed Linear Spaces
      • 7.27 Contraction Operators
      • 7.28 Fixed-Point Theorem for Contraction Operators
      • 7.29 Applications of the Fixed-Point Theorem
  • Part 2 Nonlinear Analysis
    • 8 Differential Calculus of Scalar and Vector Fields
      • 8.1 Functions from 𝐑ⁿ to R′′. Scalar and Vector Fields
      • 8.2 Open Balls and Open Sets
      • 8.3 Exercises
      • 8.4 Limits and Continuity
      • 8.5 Exercises
      • 8.6 The Derivative of a Scalar Field with Respect to a Vector
      • 8.7 Directional Derivatives and Partial Derivatives
      • 8.8 Partial Derivatives of Higher Order
      • 8.9 Exercises
      • 8.10 Directional Derivatives and Continuity
      • 8.11 The Total Derivative
      • 8.12 The Gradient of a Scalar Field
      • 8.13 A Sufficient Condition for Differentiability
      • 8.14 Exercises
      • 8.15 A Chain Rule for Derivatives of Scalar Fields
      • 8.16 Applications to Geometry. Level Sets. Tangent Planes
      • 8.17 Exercises
      • 8.18 Derivatives of Vector Fields
      • 8.19 Differentiability Implies Continuity
      • 8.20 The Chain Rule for Derivatives of Vector Fields
      • 8.21 Matrix Form of the Chain Rule
      • 8.22 Exercises
      • 8.23 Sufficient Conditions for the Equality of Mixed Partial Derivatives
    • 9 Applications of the Differential Calculus
      • 9.1 Partial Differential Equations
      • 9.2 A First-Order Partial Differential Equation with Constant Coefficients
      • 9.3 Exercises
      • 9.4 The One-Dimensional Wave Equation
      • 9.5 Exercises
      • 9.6 Derivatives of Functions Defined Implicitly
      • 9.7 Worked Examples
      • 9.8 Exercises
      • 9.9 Maxima, Minima, and Saddle Points
      • 9.10 Second-Order Taylor Formula for Scalar Fields
      • 9.11 The Nature of a Stationary Point Determined by the Eigenvalues of the Hessian Matrix
      • 9.12 Second-Derivative Test for Extrema of Functions of Two Variables
      • 9.13 Exercises
      • 9.14 Extrema with Constraints. Lagrange’s Multipliers
      • 9.15 Exercises
      • 9.16 The Extreme-Value Theorem for Continuous Scalar Fields
      • 9.17 The Small-Span Theorem for Continuous Scalar Fields (Uniform Continuity)
    • 10 Line Integrals
      • 10.1 Introduction
      • 10.2 Paths and Line Integrals
      • 10.3 Other Notations for Line Integrals
      • 10.4 Basic Properties of Line Integrals
      • 10.5 Exercises
      • 10.6 The Concept of Work as a Line Integral
      • 10.7 Line Integrals with Respect to Arc Length
      • 10.8 Further Applications of Line Integrals
      • 10.9 Exercises
      • 10.10 Open Connected Sets. Independence of the Path
      • 10.11 The Second Fundamental Theorem of Calculus for Line Integrals
      • 10.12 Applications to Mechanics
      • 10.13 Exercises
      • 10.14 The First Fundamental Theorem of Calculus for Line Integrals
      • 10.15 Necessary and Sufficient Conditions for a Vector Field to Be a Gradient
      • 10.16 Necessary Conditions for a Vector Field to Be a Gradient
      • 10.17 Special Methods for Constructing Potential Functions
      • 10.18 Exercises
      • 10.19 Applications to Exact Differential Equations of First Order
      • 10.20 Exercises
      • 10.21 Potential Functions on Convex Sets
    • 11 Multiple Integrals
      • 11.1 Introduction
      • 11.2 Partitions of Rectangles. Step Functions
      • 11.3 The Double Integral of a Step Function
      • 11.4 The Definition of the Double Integral of a Function Defined and Bounded on a Rectangle
      • 11.5 Upper and Lower Double Integrals
      • 11.6 Evaluation of a Double Integral by Repeated One-Dimensional Integration
      • 11.7 Geometric Interpretation of the Double Integral as a Volume
      • 11.8 Worked Examples
      • 11.9 Exercises
      • 11.10 Integrability of Continuous Functions
      • 11.11 Integrability of Bounded Functions with Discontinuities
      • 11.12 Double Integrals Extended over More General Regions
      • 11.13 Applications to Area and Volume
      • 11.14 Worked Examples
      • 11.15 Exercises
      • 11.16 Further Applications of Double Integrals
      • 11.17 Two Theorems of Pappus
      • 11.18 Exercises
      • 11.19 Green’s Theorem in the Plane
      • 11.20 Some Applications of Green’s Theorem
      • 11.21 A Necessary and Sufficient Condition for a Two-Dimensional Vector Field to Be a Gradient
      • 11.22 Exercises
      • 11.23 Green’s Theorem for Multiply Connected Regions
      • 11.24 The Winding Number
      • 11.25 Exercises
      • 11.26 Change of Variables in a Double Integral
      • 11.27 Special Cases of the Transformation Formula
      • 11.28 Exercises
      • 11.29 Proof of the Transformation Formula in a Special Case
      • 11.30 Proof of the Transformation Formula in the General Case
      • 11.31 Extensions to Higher Dimensions
      • 11.32 Change of Variables in an n-Fold Integral
      • 11.33 Worked Examples
    • 12 Surface Integrals
      • 12.1 Parametric Representation of a Surface
      • 12.2 The Fundamental Vector Product
      • 12.3 The Fundamental Vector Product as a Normal to the Surface
      • 12.4 Exercises
      • 12.5 Area of a Parametric Surface
      • 12.6 Exercises
      • 12.7 Surface Integrals
      • 12.8 Change of Parametric Representation
      • 12.9 Other Notations for Surface Integrals
      • 12.10 Exercises
      • 12.11 The Theorem of Stokes
      • 12.12 The Curl and Divergence of a Vector Field
      • 12.13 Exercises
      • 12.14 Further Properties of the Curl and Divergence
      • 12.15 Exercises
      • 12.16 Reconstruction of a Vector Field from Its Curl
      • 12.17 Exercises
      • 12.18 Extensions of Stokes’ Theorem
      • 12.19 The Divergence Theorem (Gauss’ Theorem)
      • 12.20 Applications of the Divergence Theorem
  • Part 3 Special Topics
    • 13 Set Functions and Elementary Probability
      • 13.1 Historical Introduction
      • 13.2 Finitely Additive Set Functions
      • 13.3 Finitely Additive Measures
      • 13.4 Exercises
      • 13.5 The Definition of Probability for Finite Sample Spaces
      • 13.6 Special Terminology Peculiar to Probability Theory
      • 13.7 Exercises
      • 13.8 Worked Examples
      • 13.9 Exercises
      • 13.10 Some Basic Principles of Combinatorial Analysis
      • 13.11 Exercises
      • 13.12 Conditional Probability
      • 13.13 Independence
      • 13.14 Exercises
      • 13.15 Compound Experiments
      • 13.16 Bernoulli Trials
      • 13.17 The Most Probable Number of Successes in 𝑛 Bernoulli Trials
      • 13.18 Exercises
      • 13.19 Countable and Uncountable Sets
      • 13.20 Exercises
      • 13.21 The Definition of Probability for Countably Infinite Sample Spaces
      • 13.22 Exercises
      • 13.23 Miscellaneous Exercises on Probability
    • 14 Calculus of Probabilities
      • 14.1 The Definition of Probability for Uncountable Sample Spaces
      • 14.2 Countability of the Set of Points with Positive Probability
      • 14.3 Random Variables
      • 14.4 Exercises
      • 14.5 Distribution Functions
      • 14.6 Discontinuities of Distribution Functions
      • 14.7 Discrete Distributions. Probability Mass Functions
      • 14.8 Exercises
      • 14.9 Continuous Distributions. Density Functions
      • 14.10 Uniform Distribution over an Interval
      • 14.11 Cauchy’s Distribution
      • 14.12 Exercises
      • 14.13 Exponential Distributions
      • 14.14 Normal Distributions
      • 14.15 Remarks on More General Distributions
      • 14.16 Exercises
      • 14.17 Distributions of Functions of Random Variables
      • 14.18 Exercises
      • 14.19 Distributions of Two-Dimensional Random Variables
      • 14.20 Two-Dimensional Discrete Distributions
      • 14.21 Two-Dimensional Continuous Distributions. Density Functions
      • 14.22 Exercises
      • 14.23 Distributions of Functions of Two Random Variables
      • 14.24 Exercises
      • 14.25 Expectation and Variance
      • 14.26 Expectation of a Function of a Random Variable
      • 14.27 Exercises
      • 14.28 Chebyshev’s Inequality
      • 14.29 Laws of Large Numbers
      • 14.30 The Central Limit Theorem of the Calculus of Probabilities
    • 15 Introduction to Numerical Analysis
      • 15.1 Historical Introduction
      • 15.2 Approximations by Polynomials
      • 15.3 Polynomial Approximation and Normed Linear Spaces
      • 15.4 Fundamental Problems in Polynomial Approximation
      • 15.5 Exercises
      • 15.6 Interpolating Polynomials
      • 15.7 Equally Spaced Interpolation Points
      • 15.8 Error Analysis in Polynomial Interpolation
      • 15.9 Exercises
      • 15.10 Newton’s Interpolation Formula
      • 15.11 Equally Spaced Interpolation Points. The Forward Difference Operator
      • 15.12 Factorial Polynomials
      • 15.13 Exercises
      • 15.14 A Minimum Problem Relative to the Max Norm
      • 15.15 Chebyshev Polynomials
      • 15.16 A Minimal Property of Chebyshev Polynomials
      • 15.17 Application to the Error Formula for Interpolation
      • 15.18 Exercises
      • 15.19 Approximate Integration. The Trapezoidal Rule
      • 15.20 Simpson’s Rule
      • 15.21 Exercises
      • 15.22 The Euler Summation Formula

Use Nagwa in conjunction with your preferred textbook. The recommended lessons from Nagwa for each section of this textbook are provided above. This alignment is not affiliated with, sponsored by, or endorsed by the publisher of the referenced textbook. Nagwa is a registered trademark of Nagwa Limited. All other trademarks and registered trademarks are the property of their respective owners.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.