Advanced Calculus

Table of Contents



1. Setting the Stage.
Euclidean Spaces and Vectors. Subsets of Euclidean Space. Limits and Continuity. Sequences. Completeness. Compactness. Connectedness. Uniform Continuity.


2. Differential Calculus.
Differentiability in One Variable. Differentiability in Several Variables. The Chain Rule. The Mean Value Theorem. Functional Relations and Implicit Functions: A First Look. Higher-Order Partial Derivatives. Taylor\'s Theorem. Critical Points. Extreme Value Problems. Vector-Valued Functions and Their Derivatives.


3. The Implicit Function Theorem and Its Applications.
The Implicit Function Theorem. Curves in the Plane. Surfaces and Curves in Space. Transformations and Coordinate Systems. Functional Dependence.


4. Integral Calculus.
Integration on the Line. Integration in Higher Dimensions. Multiple Integrals and Iterated Integrals. Change of Variables for Multiple Integrals. Functions Defined by Integrals. Improper Integrals. Improper Multiple Integrals. Lebesgue Measure and the Lebesgue Integral.


5. Line and Surface Integrals; Vector Analysis.
Arc Length and Line Integrals. Green\'s Theorem. Surface Area and Surface Integrals. Vector Derivatives. The Divergence Theorem. Some Applications to Physics. Stokes\'s Theorem. Integrating Vector Derivatives. Higher Dimensions and Differential Forms.


6. Infinite Series.
Definitions and Examples. Series with Nonnegative Terms. Absolute and Conditional Convergence. More Convergence Tests. Double Series; Products of Series.


7. Functions Defined by Series and Integrals.
Sequences and Series of Functions. Integrals and Derivatives of Sequences and Series. Power Series. The Complex Exponential and Trig Functions. Functions Defined by Improper Integrals. The Gamma Function. Stirling\'s Formula.


8. Fourier Series.
Periodic Functions and Fourier Series. Convergence of Fourier Series. Derivatives, Integrals, and Uniform Convergence. Fourier Series on Intervals. Applications to Differential Equations. The Infinite-Dimensional Geometry of Fourier Series. The Isoperimetric Inequality.

APPENDICES.
A. Summary of Linear Algebra.

Vectors. Linear Maps and Matrices. Row Operations and Echelon Forms. Determinants. Linear Independence. Subspaces; Dimension; Rank. Invertibility. Eigenvectors and Eigenvalues.
B. Some Technical Proofs.

The Heine-Borel Theorem. The Implicit Function Theorem. Approximation by Riemann Sums. Double Integrals and Iterated Integrals. Change of Variables for Multiple Integrals. Improper Multiple Integrals. Green\'s Theorem and the Divergence Theorem.
Answers to Selected Exercises.

Bibliography.

Index.