Primer of Modern Analysis

I.- 1 Applications.- 1. Tangent Lines.- 2. Derivatives.- 3. Maximum and Minimum Problems.- 4. Velocity and Acceleration.- 5. Area.- 2 Calculation of Derivatives.- 1. Limits.- 2. Limits and Derivatives.- 3. Derivatives of Sums, Products, and Quotients.- 4. Continuity.- 5. Trigonometric Functions.- 6. Composite Functions.- 7. Logarithms and Exponentials.- 3 Deeper Properties of Continuous Functions.- 1. Inverse Functions.- 2. Uniform Continuity.- 3. Maxima and Minima.- 4. The Mean-Value Theorem.- 5. Zero and Infinity.- 4 Riemann Integration.- 1. Area.- 2. Integrals.- 3. Elementary Functions.- 4. Change of Variable.- 5. Integration by Parts.- 6. Riemann Sums.- 7. Arc Length.- 8. Polar Coordinates.- 9. Volume.- 10. Improper Integrals.- 5 Taylor's Formula.- 1. Taylor's Formula.- 2. Equivalent Formulas.- 3. Local Maxima and Minima.- 6 Sequences and Series.- 1. Sequences and Series.- 2. Increasing Sequences and Positive Series.- 3. Cauchy Sequences.- 4. Sequences of Functions.- 5. Power Series.- 6. Analytic Functions.- 7. Examples.- 8. Weierstrass Approximation Theorem.- II.- 7 Metric Spaces.- 1. The spaceRn.- 2. Absolute Value in Rn.- 3. Metric Spaces.- 4. Function Spaces.- 5. Equivalent Metrics.- 6. Open and Closed Sets.- 7. Connected Spaces.- 8. Composite Functions and Subsequences.- 9. Compact Spaces.- 10. Equivalence of Absolute Values on Rn.- 11. Products.- 12. Stone-Weierstrass Approximation Theorem.- 8 Functions From R1to Rn.- 1. Lines, Half-lines, and Directions.- 2. Derivatives and Integrals.- 3. Tangent Lines, Velocity, and Acceleration.- 4. Geometric Models of R".- 5. Missiles, Moons, and so on.- 6. Arc Length.- 9 Algebra and Geometry in Rn.- 1. Subspaces.- 2. Bases.- 3. Orthonormal Bases.- 4. Linear Transformations.- 5. Sums and Products.- 6. Null Space and Range.- 7. Matrices and Linear Equations.- 8. Continuity of Linear Transformations.- 9. Self-adjoint Transformations.- 10. Orthogonal Transformations.- 11. Determinants.- 10 Linear Approximation.- 1. Directional Derivatives and Partial Derivatives.- 2. The Differential.- 3. Existence of the Differential.- 4. Composite Functions.- 5. The Mean-Value Theorem.- 6. A Fixed-Point Theorem.- 7. The Inverse-Function Theorem.- 8. The Implicit-Function Theorem.- 11 Surfaces.- 1. Algebraic Curves.- 2. Manifolds.- 3. Tangent Spaces.- 4. Functions on Manifolds.- 5. Quadratic Forms and Quadric Surfaces.- 12 Higher Derivatives.- 1. Second Derivatives.- 2. Higher Derivatives.- 3. The Inverse-and Implicit-Function Theorems.- 4. Taylor's Formula.- 5. Local Maxima and Minima.- III.- 13 Integration.- 1. Introduction.- 2. Lebesgue Measure.- 3. Outer Measures.- 4. Measurability in RRn.- 5. Measurable Functions.- 6. Definition of the Integral.- 7. Convergence Theorems.- 8. Integrable Functions.- 9. Product Measures.- 10. Functions Defined by Integrals.- 11. Convolution.- 12. Approximation Theorems.- 13. Multiple Series.- 14. Regular Values and Sard's Theorem.- 14 Differentiation.- 1. Regular Borel Measures.- 2. Differentiability Theorems.- 3. Integration of Derivatives.- 4. Change of Variable.- 5. Differentiability of Lipschitz Functions.- 15 Surface Area.- 1. Area Measures.- 2. Parametric-SurfacesIntroductory Remarks.- 3. The Jacobian.- 4. Absolute Continuity.- 5. Variation.- 6. The Jacobian Formula for Surface Area.- 7. Examples.- 8. Polar Coordinates.- 16 The Brouwer Degree.- 1. Introduction.- 2. The Degree for C?Functions.- 3. The Degree for Continuous Functions.- 4. Some Applications of the Degree.- 5. Change of Variable Revisited.- 17 Extensions of Differentiable Functions.- 1. Introduction.- 2. Reflection Across Hyperplanes.- 3. Regularized Distance.- 4. Reflection Across Lipschitz Graphs.- 5. Reflection of Holder Functions.- 6. Reflection of Sobolev Functions.- 7. Extension from Lipschitz Graph Domains.