Calculus and Linear Algebra. Vol. 2: Vector Spaces, Many-Variable Calculus, and Differential Equations
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Contents
- Frontmatter
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CHAPTER 9 VECTOR SPACES
- 9-1 The Concept of a Vector Space 641
- 9-2 Subspaces 647
- 9-3 Intersection of Subspaces 652
- 9-4 Addition of Subsets 655
- 9-5 Linear Varieties 661
- 9-6 Span of a Set 665
- 9-7 Bases, Linear Independence 666
- 9-8 Dimension 672
- 9-9 Dimension of Subspaces and of Linear Varieties 674
- ‡9-10 Proofs of Theorems on Dimension 676
- 9-11 Linear Mappings 681
- 9-12 Range of a Linear Mapping 687
- 9-13 Kernal of a Linear Mapping 688
- 9-14 Rank and Nullity of a Linear Mapping 691
- ‡9-15 Proofs of Two Theorems 694
- 9-16 Addition of Linear Mappings, Scalar Multiples of Linear Mappings 696
- 9-17 Composition of Linear Mappings 698
- 9-18 Inverse of a Linear Mapping 700
- 9-19 Linear Transformation on a Vector Space 703
- 9-20 Polynomials in a Linear Transformation 705
- 9-21 Nonsingular Linear Transformations 709
- 9-22 The Minimal Polynomial of a Linear Transformation 712
- 9-23 Eigenvectors and Eigenvalues 714
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CHAPTER 10 MATRICES AND DETERMINANTS
- 10-1 Matrices 718
- 10-2 Matrices and Linear Mappings of Vn into Vm 719
- 10-3 Matrices as Linear Mappings 723
- 10-4 Kernel, Range, Nullity, and Rank of a Matrix 724
- 10-5 Identity Matrix, Scalar Matrix, Zero Matrix, Complex Matrices 727
- 10-6 Linear Equations 730
- 10-7 Addition of Matrices, Scalar Times Matrix 740
- 10-8 Multiplication of Matrices 742
- 10-9 The Transpose 745
- 10-10 Partitioning of a Matrix 748
- 10-11 The Algebra of Square Matrices 750
- 10-12 Nonsingular Matrices 756
- 10-13 Determinants 760
- ‡10-14 Proofs of Theorems on Determinants 771
- ‡10-15 Further Remarks on Determinants 776
- †10-16 The Method of Elimination 781
- †10-17 Matrices of Functions 788
- †10-18 Eigenvalues, Eigenvectors, Characteristic Polynomial of a Matrix 790
- ‡10-19 Matrix Representations of a Linear Mapping 795
- ‡10-20 Jordan Matrices 796
- ‡10-21 Similar Matrices 799
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CHAPTER 11 LINEAR EUCLIDEAN GEOMETRY
- Introduction 804
- 11-1 Inner Product and Norm in V₃ 805
- 11-2 Unit Vectors, Angle Between Vectors 807
- ‡11-3 Euclidean Vector Space of Dimension n 808
- 11-4 Points, Vectors, Distance, Lines in 3-Dimensional Euclidean Space R³ 811
- ‡11-5 Lines in n-Dimensional Euclidean Space 817
- 11-6 The Cross Product (Vector Product) 819
- 11-7 Triple Products 824
- 11-8 Application of the Cross Product to Lines in Space 826
- ‡11-9 The Cross Product in Vn 828
- 11-10 Planes in R³ 831
- 11-11 Relations between Lines and Planes 838
- 11-12 Relations between Two Planes 840
- ‡11-13 Hyperplanes and Linear Manifolds in Rⁿ 842
- 11-14 Other Cartesian Coordinate Systems in R³ 844
- 11-15 Lengths, Areas, and Volumes in R³ 848
- ‡11-16 New Coordinates and Volume in Rⁿ 854
- 11-17 Linear Mappings of R³ into R³ 857
- ‡11-18 Linear Mappings of Rⁿ into Rm 860
- 11-19 Surfaces in R³ 862
- 11-20 Cylindrical and Spherical Coordinates 865
- 11-21 Change of Coordinates in R³ 868
- ‡11-22 Change of Coordinates in Rⁿ 873
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CHAPTER 12 DIFFERENTIAL CALCULUS OF FUNCTIONS OF SEVERAL VARIABLES
- Introduction 875
- 12-1 Sets in the Plane 876
- 12-2 Functions of Two Variables 878
- 12-3 Functions of Three or More Variables 883
- 12-4 Vector Functions 884
- 12-5 Matrix Functions 886
- 12-6 Operations on Functions 887
- 12-7 Limits and Continuity 889
- 12-8 Partial Derivatives 897
- 12-9 The Differential 902
- 12-10 Chain Rules 908
- 12-11 The Directional Derivative 913
- 12-12 Differential of a Vector Function, the Jacobian Matrix 918
- 12-13 The General Chain Rule 922
- 12-14 Implicit Functions 926
- ‡12-15 Implicit Function Theorem 935
- 12-16 Inverse Functions 939
- 12-17 Curves in Space 945
- 12-18 Surfaces in Space 948
- 12-19 Partial Derivatives of Higher Order 954
- ‡12-20 Proof of Theorem on Mixed Partial Derivatives 957
- 12-21 Taylor's Formula 960
- 12-22 Maxima and Minima of Functions of Two Variables 966
- ‡12-23 Lagrange Multipliers 974
- ‡12-24 Proof of Theorem on Local Maxima and Minima 976
- ‡12-25 Some Deeper Results on Continuity 980
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CHAPTER 13 INTEGRAL CALCULUS OF FUNCTIONS OF SEVERAL VARIABLES
- 13-1 The Double Integral 988
- 13-2 Theory of the Double Integral 997
- ‡13-3 Proof that the Double Integral can be Represented as a Limit 1005
- 13-4 Double Integrals in Polar Coordinates 1010
- ‡13-5 Other Curvilinear Coordinates 1013
- 13-6 Triple Integrals 1017
- 13-7 Triple Integrals in Cylindrical and Spherical Coordinates 1023
- 13-8 Further Properties of Multiple Integrals 1031
- 13-9 Surface Area 1037
- 13-10 Other Applications of Multiple Integrals 1041
- 13-11 Line Integrals 1049
- 13-12 Green's Theorem 1060
- 13-13 Curl and Divergence, Vector Form of Green's Theorem 1062
- 13-14 Exact Differentials and Independence of Path 1073
- 13-15 The Divergence Theorem and Stokes' Theorem in Space 1081
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CHAPTER 14 ORDINARY DIFFERENTIAL EQUATIONS
- 14-1 Basic Concepts 1088
- 14-2 Graphical Method and Method of Step-by-Step Integration 1096
- 14-3 Exact First-Order Equations 1100
- 14-4 Equations with Variables Separable and Equations of Form y' = g(y/x) 403
- 14-5 The Linear Equation of First Order 1111
- 14-6 Linear Differential Equations of Order n 1117
- 14-7 Variation of Parameters 1124
- 14-8 Complex-Valued Solutions of Linear Differential Equations 1126
- 14-9 Homogeneous Linear Differential Equations with Constant Coefficients 1129
- ‡14-10 Linear Independence of Solutions of the Homogeneous Linear Equation with Constant Coefficients 1132
- 14-11 Nonhomogeneous Linear Differential Equations with Constant Coefficients 1134
- 14-12 Applications of Linear Differential Equations 1137
- 14-13 Vibrations of a Mass-Spring System 1142
- 14-14 Simultaneous Linear Differential Equations 1148
- 14-15 Solutions Satisfying Initial Conditions, Variation of Parameters 1154
- 14-16 Complex-Valued Solutions of Systems of Linear Differential Equations 1159
- 14-17 Homogeneous Linear Systems with Constant Coefficients 1162
- 14-18 Nonhomogeneous Linear Systems with Constant Coefficients: Stability 1166
- 14-19 Method of Elimination 1171
- ‡14-20 Application of Exponential Function of a Matrix 1174
- 14-21 Autonomous Linear Systems of Order Two 1179
- 14-22 Power Series Solutions 1187
- 14-23 Numerical Solutions of Differential Equations 1193
- ANSWERS TO SELECTED PROBLEMS A-45
- INDEX I-7
