An introduction to multivariable mathematics
The text is designed for use in a 40 lecture introductory course covering linear algebra, multivariable differential calculus, and an introduction to real analysis. The core material of the book is arranged to allow for the main introductory material on linear algebra, including basic vector space t...
| Main Author: | |
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| Format: | Electronic |
| Language: | English |
| Published: |
San Rafael, Calif. (1537 Fourth Street, San Rafael, CA 94901 USA) :
Morgan & Claypool Publishers,
c2008.
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| Series: | Synthesis lectures on mathematics and statistics (Online) ;
#3. |
| Subjects: | |
| Online Access: | View fulltext via EzAccess |
Table of Contents:
- Linear algebra
- Vectors in Rn
- Dot product and angle between vectors in Rn
- Subspaces and linear dependence of vectors
- Gaussian elimination and the linear dependence lemma
- The basis theorem
- Matrices
- Rank and the rank-nullity theorem
- Orthogonal complements and orthogonal projection
- Row echelon form of a matrix
- Inhomogeneous systems
- Analysis in Rn
- Open and closed sets in Euclidean space
- Bolzano-Weierstrass, limits and continuity in Rn
- Differentiability
- Directional derivatives, partial derivatives, and gradient
- Chain rule
- Higher-order partial derivatives
- Second derivative test for extrema of multivariable function
- Curves in Rn
- Submanifolds of Rn and tangential gradients
- More linear algebra
- Permutations
- Determinants
- Inverse of a square matrix
- Computing the inverse
- Orthonormal basis and Gram-Schmidt
- Matrix representations of linear transformations
- Eigenvalues and the spectral theorem
- More analysis in Rn
- Contraction mapping principle
- Inverse function theorem
- Implicit function theorem
- Introductory lectures on real analysis
- Lecture 1: The real numbers
- Lecture 2: Sequences of real numbers and the Bolzano-Weierstrass theorem
- Lecture 3: Continuous functions
- Lecture 4: Series of real numbers
- Lecture 5: Power series
- Lecture 6: Taylor series representations
- Lecture 7: Complex series, products of series, and complex exponential series
- Lecture 8: Fourier series
- Lecture 9: Pointwise convergence of trigonometric Fourier series.