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Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds



Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds

Theodore Shifrin

ISBN: 978-0-471-52638-4 January 2004 512 Pages


Multivariable Mathematics combines linear algebra and multivariable calculus in a rigorous approach. The material is integrated to emphasize the role of linearity in all of calculus and the recurring theme of implicit versus explicit that persists in linear algebra and analysis.  In the text, the author addresses all of the standard computational material found in the usual linear algebra and multivariable calculus courses, and more, interweaving the material as effectively as possible and also including complete proofs.  By emphasizing the theoretical aspects and reviewing the linear algebra material quickly, the book can also be used as a text for an advanced calculus or multivariable analysis course culminating in a treatment of manifolds, differential forms, and the generalized Stokes’s Theorem.

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Chapter 1. Vectors and Matrices.

1.1 Vectors in Rn..

1.2 Dot Product.

1.3 Subspaces of Rn.

1.4 Linear Transformations and Matrix Algebra.

1.5 Introduction to Determinates and the Cross Product.

Chapter 2. Functions, Limits, and Continuity.

2.1. Scalar- and Vector-Valued Functions.

2.2. A Bit of Topology in Rn.

2.3. Limits and Continuity.

Chapter 3. The Derivative.

3.1. Partial Derivatives and Directional Derivatives.

3.2. Differentiability.

3.3. Differentiation Rules.

3.4. The Gradient.

3.5. Curves.

3.6. Higher-Order Partial Derivatives.

Chapter  4. Implicit and Explicit Solutions of Linear Systems.

4.1. Gaussian Elimination and the Theory of Linear Systems.

4.2. Elementary Matrices and Calculating Inverse Matrices.

4.3. Linear Independence, Basis, and Dimension.

4.4. The Four Fundamental Subspaces.

4.5. The Nonlinear Case: Introduction to Manifolds.

Chapter 5. Extremum Problems.

5.1. Compactness and the Maximum Value Theorem.

5.2. Maximum/Minimum Problems.

5.3. Quadratic Forms and the Second Derivative Test.

5.4. Lagrange Multipliers.

5.5. Projections, Least Squares, and Inner Product Spaces. 

Chapter 6. Solving Nonlinear Problems.

6.1. The Contraction Mapping Principle.

6.2. The Inverse and Implicit Function Theorems.

6.3. Manifolds Revisited.

Chapter 7. Integration.

7.1. Multiple Integrals.

7.2. Iterated Integrals and Fubini’s Theorem.

7.3. Polar, Cylindrical, and Spherical Coordinates.

7.4. Physical Applications.

7.5. Determinants and n-Dimensional Volume.

7.6. Change of Variables Theorem.

Chapter 8. Differential Forms and Integration on Manifolds.

8.1. Motivation.

8.2. Differential Forms.

8.3. Line Integrals and Green’s Theorem.

8.4. Surface Integrals and Flux.

8.5. Stokes’s Theorem.

8.6. Applications to Physics.

8.7. Applications to Topology.

9.  Eigenvalues, Eigenvectors, and Applications.

9.1. Linear Transformations and Change of Basis.

9.2. Eigenvalues, Eigenvectors, and Diagonalizability.

9.3. Difference Equations and Ordinary Differential Equations.

9.4. The Spectral Theorem.

Glossary of Notations and Results from Single-Variable Calculus.

For Further Reading.

Answers to Selected Exercises.


Key Features:
  •  Contains plenty of examples, clear proofs, and significant motivation for the crucial concepts.
  • Includes numerous exercises of varying levels of difficulty, both computational and more proof-oriented.
  • Exercises are arranged in order of increasing difficulty.