Calculus and Linear Algebra. Vol. 2: Vector Spaces, ManyVariable Calculus, and Differential Equations
Skip other details (including permanent urls, DOI, citation information) :
: Ann Arbor, MI: Michigan Publishing, University of Michigan Library, 2007.
This work is protected by copyright and may be linked to without seeking permission. Permission must be received for subsequent distribution in print or electronically. Please contact mpubhelp@umich.edu for more information. :
For more information, read Michigan Publishing's access and usage policy.
Contents
 Frontmatter

CHAPTER 9 VECTOR SPACES
 91 The Concept of a Vector Space 641
 92 Subspaces 647
 93 Intersection of Subspaces 652
 94 Addition of Subsets 655
 95 Linear Varieties 661
 96 Span of a Set 665
 97 Bases, Linear Independence 666
 98 Dimension 672
 99 Dimension of Subspaces and of Linear Varieties 674
 ‡910 Proofs of Theorems on Dimension 676
 911 Linear Mappings 681
 912 Range of a Linear Mapping 687
 913 Kernal of a Linear Mapping 688
 914 Rank and Nullity of a Linear Mapping 691
 ‡915 Proofs of Two Theorems 694
 916 Addition of Linear Mappings, Scalar Multiples of Linear Mappings 696
 917 Composition of Linear Mappings 698
 918 Inverse of a Linear Mapping 700
 919 Linear Transformation on a Vector Space 703
 920 Polynomials in a Linear Transformation 705
 921 Nonsingular Linear Transformations 709
 922 The Minimal Polynomial of a Linear Transformation 712
 923 Eigenvectors and Eigenvalues 714

CHAPTER 10 MATRICES AND DETERMINANTS
 101 Matrices 718
 102 Matrices and Linear Mappings of V_{n} into V_{m} 719
 103 Matrices as Linear Mappings 723
 104 Kernel, Range, Nullity, and Rank of a Matrix 724
 105 Identity Matrix, Scalar Matrix, Zero Matrix, Complex Matrices 727
 106 Linear Equations 730
 107 Addition of Matrices, Scalar Times Matrix 740
 108 Multiplication of Matrices 742
 109 The Transpose 745
 1010 Partitioning of a Matrix 748
 1011 The Algebra of Square Matrices 750
 1012 Nonsingular Matrices 756
 1013 Determinants 760
 ‡1014 Proofs of Theorems on Determinants 771
 ‡1015 Further Remarks on Determinants 776
 †1016 The Method of Elimination 781
 †1017 Matrices of Functions 788
 †1018 Eigenvalues, Eigenvectors, Characteristic Polynomial of a Matrix 790
 ‡1019 Matrix Representations of a Linear Mapping 795
 ‡1020 Jordan Matrices 796
 ‡1021 Similar Matrices 799

CHAPTER 11 LINEAR EUCLIDEAN GEOMETRY
 Introduction 804
 111 Inner Product and Norm in V₃ 805
 112 Unit Vectors, Angle Between Vectors 807
 ‡113 Euclidean Vector Space of Dimension n 808
 114 Points, Vectors, Distance, Lines in 3Dimensional Euclidean Space R³ 811
 ‡115 Lines in nDimensional Euclidean Space 817
 116 The Cross Product (Vector Product) 819
 117 Triple Products 824
 118 Application of the Cross Product to Lines in Space 826
 ‡119 The Cross Product in V_{n} 828
 1110 Planes in R³ 831
 1111 Relations between Lines and Planes 838
 1112 Relations between Two Planes 840
 ‡1113 Hyperplanes and Linear Manifolds in Rⁿ 842
 1114 Other Cartesian Coordinate Systems in R³ 844
 1115 Lengths, Areas, and Volumes in R³ 848
 ‡1116 New Coordinates and Volume in Rⁿ 854
 1117 Linear Mappings of R³ into R³ 857
 ‡1118 Linear Mappings of Rⁿ into R^{m} 860
 1119 Surfaces in R³ 862
 1120 Cylindrical and Spherical Coordinates 865
 1121 Change of Coordinates in R³ 868
 ‡1122 Change of Coordinates in Rⁿ 873

CHAPTER 12 DIFFERENTIAL CALCULUS OF FUNCTIONS OF SEVERAL VARIABLES
 Introduction 875
 121 Sets in the Plane 876
 122 Functions of Two Variables 878
 123 Functions of Three or More Variables 883
 124 Vector Functions 884
 125 Matrix Functions 886
 126 Operations on Functions 887
 127 Limits and Continuity 889
 128 Partial Derivatives 897
 129 The Differential 902
 1210 Chain Rules 908
 1211 The Directional Derivative 913
 1212 Differential of a Vector Function, the Jacobian Matrix 918
 1213 The General Chain Rule 922
 1214 Implicit Functions 926
 ‡1215 Implicit Function Theorem 935
 1216 Inverse Functions 939
 1217 Curves in Space 945
 1218 Surfaces in Space 948
 1219 Partial Derivatives of Higher Order 954
 ‡1220 Proof of Theorem on Mixed Partial Derivatives 957
 1221 Taylor's Formula 960
 1222 Maxima and Minima of Functions of Two Variables 966
 ‡1223 Lagrange Multipliers 974
 ‡1224 Proof of Theorem on Local Maxima and Minima 976
 ‡1225 Some Deeper Results on Continuity 980

CHAPTER 13 INTEGRAL CALCULUS OF FUNCTIONS OF SEVERAL VARIABLES
 131 The Double Integral 988
 132 Theory of the Double Integral 997
 ‡133 Proof that the Double Integral can be Represented as a Limit 1005
 134 Double Integrals in Polar Coordinates 1010
 ‡135 Other Curvilinear Coordinates 1013
 136 Triple Integrals 1017
 137 Triple Integrals in Cylindrical and Spherical Coordinates 1023
 138 Further Properties of Multiple Integrals 1031
 139 Surface Area 1037
 1310 Other Applications of Multiple Integrals 1041
 1311 Line Integrals 1049
 1312 Green's Theorem 1060
 1313 Curl and Divergence, Vector Form of Green's Theorem 1062
 1314 Exact Differentials and Independence of Path 1073
 1315 The Divergence Theorem and Stokes' Theorem in Space 1081

CHAPTER 14 ORDINARY DIFFERENTIAL EQUATIONS
 141 Basic Concepts 1088
 142 Graphical Method and Method of StepbyStep Integration 1096
 143 Exact FirstOrder Equations 1100
 144 Equations with Variables Separable and Equations of Form y' = g(y/x) 403
 145 The Linear Equation of First Order 1111
 146 Linear Differential Equations of Order n 1117
 147 Variation of Parameters 1124
 148 ComplexValued Solutions of Linear Differential Equations 1126
 149 Homogeneous Linear Differential Equations with Constant Coefficients 1129
 ‡1410 Linear Independence of Solutions of the Homogeneous Linear Equation with Constant Coefficients 1132
 1411 Nonhomogeneous Linear Differential Equations with Constant Coefficients 1134
 1412 Applications of Linear Differential Equations 1137
 1413 Vibrations of a MassSpring System 1142
 1414 Simultaneous Linear Differential Equations 1148
 1415 Solutions Satisfying Initial Conditions, Variation of Parameters 1154
 1416 ComplexValued Solutions of Systems of Linear Differential Equations 1159
 1417 Homogeneous Linear Systems with Constant Coefficients 1162
 1418 Nonhomogeneous Linear Systems with Constant Coefficients: Stability 1166
 1419 Method of Elimination 1171
 ‡1420 Application of Exponential Function of a Matrix 1174
 1421 Autonomous Linear Systems of Order Two 1179
 1422 Power Series Solutions 1187
 1423 Numerical Solutions of Differential Equations 1193
 ANSWERS TO SELECTED PROBLEMS A45
 INDEX I7