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ISBN13: 978-0471526384

ISBN10: 047152638X

Edition: 05

Copyright: 2005

Publisher: John Wiley & Sons, Inc.

Published: 2005

International: No

ISBN10: 047152638X

Edition: 05

Copyright: 2005

Publisher: John Wiley & Sons, Inc.

Published: 2005

International: No

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**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.

**1. Vectors and Matrices. **

1.1Vectors in Rn..

1.2Dot Product.

1.3Subspaces of Rn.

1.4Linear Transformations and Matrix Algebra.

1.5Introduction to Determinates and the Cross Product.

**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.

**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.

**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.

**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.

**6.Solving Nonlinear Problems. **

6.1. The Contraction Mapping Principle.

6.2. The Inverse and Implicit Function Theorems.

6.3. Manifolds Revisited.

**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.

**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.

Index.

ISBN10: 047152638X

Edition: 05

Copyright: 2005

Publisher: John Wiley & Sons, Inc.

Published: 2005

International: No

**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.

Table of Contents

**1. Vectors and Matrices. **

1.1Vectors in Rn..

1.2Dot Product.

1.3Subspaces of Rn.

1.4Linear Transformations and Matrix Algebra.

1.5Introduction to Determinates and the Cross Product.

**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.

**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.

**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.

**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.

**6.Solving Nonlinear Problems. **

6.1. The Contraction Mapping Principle.

6.2. The Inverse and Implicit Function Theorems.

6.3. Manifolds Revisited.

**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.

**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.

Index.

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