Advanced Calculus

Table of Contents



1. Vectors and Matrices.
Introduction.
Vectors in Space.
Linear Independence
Lines and Planes.
Determinants.
Simultaneous Linear Equations.
Matrices.
Addition of Matrices
Scalar Times Matrix.
Multiplication of Matrices.
Inverse of a Square Matrix.
Gaussian Elimination.
*Eigenvalues of a Square Matrix.
*The Transpose.
*Orthogonal Matrices.
Analytic Geometry and Vectors n-Dimensional Space.
*Axioms for Vn.
Linear Mappings.
*Subspaces
Rank of a Matrix.
*Other Vector Spaces.


2. Differential Calculus of Functions of Several Variables.
Functions of Several Variables.
Domains and Regions.
Functional Notation Level Curves and Level Surfaces.
Limits and Continuity.
Partial Derivatives.
Total Differential Fundamental Lemma.
Differential of Functions of n Variables
The Jacobian Matrix.
Derivatives and Differentials of Composite Functions.
The General Chain Rule.
Implicit Functions.
*Proof of a Case of the Implicit Function Theorem.
Inverse Functions
Curvilinear Coordinates.
Geometrical Applications.
The Directional Derivative.
Partial Derivatives of Higher Order.
Higher Derivatives of Composite Functions.
The Laplacian in Polar, Cylindrical, and Spherical Coordinates.
Higher Derivatives of Implicit Functions.
Mixima and Minima of Functions of Several Variables.
*Extrema for Functions with Side Conditions
Lagrange Multipliers.
*Maxima and Minima of Quadratic Forms on the Unit Sphere.
*Functional Dependence.
*Real Variable Theory
Theorem on Maximum and Minimum.


3. Vector Differential Calculus.
Introduction.
Vector Fields and Scalar Fields.
The Gradient Field.
The Divergence of a Vector Field.
The Curl of a Vector Field.
Combined Operations.
*Curvilinear Coordinates in Space.
Orthogonal Coordinates.
*Vector Operations in Orthogonal Curvilinear Coordinates.
*Tensors.
*Tensors on a Surface or Hypersurface.
*Alternating Tensors.
Exterior Product.


4. Integral Calculus of Functions of Several Variables.
The Definite Integral.
Numerical Evaluation of Indefinite Integrals.
Elliptic Integrals.
Double Integrals.
Triple Integrals and Multiple Integrals in General.
Integrals of Vector Functions.
Change of Variables in Integrals.
Arc Length and Surface Area.
Improper Multiple Integrals.
Integrals depending on a Parameter—Leibnitz\'s Rule.
*Uniform Continuity.
Existence of the Riemann Integral.
*Theory of Double Integrals.


5. Vector Integral Calculus.
Two-Dimensional Theory.
Introduction.
Line Integrals in the Plane.
Integrals with Respect to Arc Length.
Basic Properties of Line Integrals.
Line Integrals as Integrals of Vectors.
Green\'s Theorem.
Independence of Path.
Simply Connected Domains.
Extension of Results to Multiply Connected Domains.
Three-Dimensional Theory and Applications.
Line Integrals in Space.
Surfaces in Space.
Orientability.
Surface Integrals.
The Divergence Theorem.
Stokes\'s Theorem.
Integrals Independent of Path.
Irrotational and Solenoidal Fields.
*Change of Variables in a Multiple Integral.
*Physical Applications.
*Potential Theory in the Plane.
*Green\'s Third Identity.
*Potential Theory in Space.
*Differential Forms.
*Change of Variables in an m-Form and General Stokes\'s Theorem.
*Tensor Aspects of Differential Forms.
*Tensors and Differential Forms without Coordinates.


6. Infinite Series.
Introduction.
Infinite Sequences.
Upper and Lower Limits.
Further Properties of Sequences.
Infinite Series.
Tests for Convergence and Divergence.
Examples and Applications of Tests for Convergence and Divergence.
*Extended Ratio Test and Root Test.
*Computation with Series—Estimate of Error.
Operations on Series.
Sequences and Series of Functions.
Uniform Convergence.
Weierstrass M-Test for Uniform Convergence.
Properties of Uniformly Convergent Series and Sequences.
Power Series.
Taylor and MacLaurin Series.
Taylor\'s Formula with Remainder.
Further Operations on Power Series.
*Sequences and Series of Complex Numbers.
*Sequences and Series of Functions of Several Variables.
*Taylor\'s Formula for Functions of Several Variables.
*Improper Integrals Versus Infinite Series.
*Improper Integrals Depending on a Parameter—Uniform Convergence.
*Principal Value of Improper Integrals.
*Laplace Transformation.
G-Function and B-Function.
*Convergence of Improper Multiple Integrals.


7. Fourier Series and Orthogonal Functions.
Trigonometric Series.
Fourier Series.
Convergence of Fourier Series.
Examples—Minimizing of Square Error.
Generalizations.
Fourier Cosine Series.
Fourier Sine Series.
Remarks on Applications of Fourier Series.
Uniqueness Theorem.
Proof of Fundamental Theorem for Continuous, Periodic, and Piecewise Very Smooth Functions.
Proof of Fundamental Theorem.
Orthogonal Functions.
*Fourier Series of Orthogonal Functions.
Completeness.
*Sufficient Conditions for Completeness.
*Integration and Differentiation of Fourier Series.
*Fourier-Legendre Series.
*Fourier-Bessel Series.
*Orthogonal Systems of Functions of Several Variables.
*Complex Form of Fourier Series.
*Fourier Integral.
*The Laplace Transform as a Special Case of the Fourier Transform.
General Functions.


8. Functions of a Complex Variable.
Complex Functions.
Complex-Valued Functions of a Real Variable.
Complex-Valued Functions of a Complex Variable.
Limits and Continuity.
Derivatives and Differentials.
Integrals.
Analytic Functions.
Cauchy-Riemann Equations.
The Functions log z, az, za, sin-1 z, cos-1 z.
Integrals of Analytic Functions.
Cauchy Integral Theorem.
Cauchy\'s Integral Formula.
Power Series as Analytic Functions.
Power Series Expansion of General Analytic Function.
Power Series in Positive and Negative Powers.
Laurent Expansion.
Isolated Singularities of an Analytic Function.
Zeros and Poles.
The Complex Number à.
Residues.
Residue at Infinity.
Logarithmic Residues.
Argument Principle.
Partial Fraction Expansion of Rational Functions.
Application of Residues to Evaluation of Real Integrals.
Definition of Conformal Mapping.
Examples of Conformal Mapping.
Applications of Conformal Mapping.
The Dirichlet Problem.
Dirichlet Problem for the Half-Plane.
Conformal Mapping in Hydrodynamics.
Applications of Conformal Mapping in the Theory of Elasticity.
Further Applications of Conformal Mapping.
General Formulas for One-to-One Mapping.
Schwarz-Christoffel Transformation.


9. Ordinary Differential Equations.
Differential Equations.
Solutions.
The Basic Problems.
Existence Theorem.
Linear Differential Equations.
Systems of Differential Equations.
Linear Systems.
Linear Systems with Constant Coefficients.
A Class of Vibration Problems.
Solution of Differential Equations by Taylor Series.
The Existence and Uniqueness Theorem.


10. Partial Differential Equations.
Introduction.
Review of Equation for Forced Vibrations of a Spring.
Case of Two Particles.
Case of n Particles.
Continuous Medium.
Fundamental Partial Differential Equation.
Classification of Partial Differential Equations.
Basic Problems.
The Wave Equation in One Dimension.
Harmonic Motion.
Properties of Solutions of the Wave Equation.
The One-Dimensional Heat Equation.
Exponential Decay.
Properties of Solutions of the Heat Equation.
Equilibrium and Approach to Equilibrium.
Forced Motion.
Equations with Variable Coefficients.
Sturm-Liouville Problems.
Equations in Two and Three Dimensions.
Separation of Variables.
Unbounded Regions.
Continuous Spectrum.
Numerical Methods.
Variational Methods.
Partial Differential Equations and Integral Equations.