Primer of Modern Analysis

1. Front Matter

   Pages i-xv

   PDF

2. Part I

   1. Front Matter

      Pages 1-1

      PDF

   2. Applications

      • Kennan T. Smith

      Pages 3-14

   3. Calculation of Derivatives

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      Pages 15-33

   4. Deeper Properties of Continuous Functions

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      Pages 34-49

   5. Riemann Integration

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      Pages 50-79

   6. Taylor’s Formula

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      Pages 80-88

   7. Sequences and Series

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      Pages 89-119

3. Part II

   1. Front Matter

      Pages 121-121

      PDF

   2. Metric Spaces

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      Pages 123-157

   3. Functions from R1 to Rn

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      Pages 158-177

   4. Algebra and Geometry in Rn

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      Pages 178-222

   5. Linear Approximation

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      Pages 223-248

   6. Surfaces

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      Pages 249-277

   7. Higher Derivatives

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      Pages 278-288

4. Part III

   1. Front Matter

      Pages 289-289

      PDF

   2. Integration

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      Pages 291-347

   3. Differentiation

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      Pages 348-370

   4. Surface Area

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      Pages 371-395

   5. The Brouwer Degree

      • Kennan T. Smith

      Pages 396-415

This book discusses some of the first principles of modern analysis. I t can be used for courses at several levels, depending upon the background and ability of the students. It was written on the premise that today's good students have unexpected enthusiasm and nerve. When hard work is put to them, they work harder and ask for more. The honors course (at the University of Wisconsin) which inspired this book was, I think, more fun than the book itself. And better. But then there is acting in teaching, and a typewriter is a poor substitute for an audience. The spontaneous, creative disorder that characterizes an exciting course becomes silly in a book. To write, one must cut and dry. Yet, I hope enough of the spontaneity, enough of the spirit of that course, is left to enable those using the book to create exciting courses of their own. Exercises in this book are not designed for drill. They are designed to clarify the meanings of the theorems, to force an understanding of the proofs, and to call attention to points in a proof that might otherwise be overlooked. The exercises, therefore, are a real part of the theory, not a collection of side issues, and as such nearly all of them are to be done. Some drill is, of course, necessary, particularly in the calculation of integrals.

• Algebra
• Analysis
• Calculation
• Derivative
• Manifold
• Maximum
• Minimum
• calculus
• differential equation
• fixed-point theorem
• function
• geometry
• logarithm
• measure
• theorem

• Mathematics Department, Oregon State University, Corvallis, USA

  Kennan T. Smith

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