Calculus and Linear Algebra. Vol. 2: Vector Spaces, ManyVariable Calculus, and Differential Equations
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Contents
 Frontmatter

CHAPTER 9 VECTOR SPACES
 91 The Concept of a Vector Space641
 92 Subspaces647
 93 Intersection of Subspaces652
 94 Addition of Subsets655
 95 Linear Varieties661
 96 Span of a Set665
 97 Bases, Linear Independence666
 98 Dimension672
 99 Dimension of Subspaces and of Linear Varieties674
 ‡910 Proofs of Theorems on Dimension676
 911 Linear Mappings681
 912 Range of a Linear Mapping687
 913 Kernal of a Linear Mapping688
 914 Rank and Nullity of a Linear Mapping691
 ‡915 Proofs of Two Theorems694
 916 Addition of Linear Mappings, Scalar Multiples of Linear Mappings696
 917 Composition of Linear Mappings698
 918 Inverse of a Linear Mapping700
 919 Linear Transformation on a Vector Space703
 920 Polynomials in a Linear Transformation705
 921 Nonsingular Linear Transformations709
 922 The Minimal Polynomial of a Linear Transformation712
 923 Eigenvectors and Eigenvalues714

CHAPTER 10 MATRICES AND DETERMINANTS
 101 Matrices718
 102 Matrices and Linear Mappings of V_{n} into V_{m}719
 103 Matrices as Linear Mappings723
 104 Kernel, Range, Nullity, and Rank of a Matrix724
 105 Identity Matrix, Scalar Matrix, Zero Matrix, Complex Matrices727
 106 Linear Equations730
 107 Addition of Matrices, Scalar Times Matrix740
 108 Multiplication of Matrices742
 109 The Transpose745
 1010 Partitioning of a Matrix748
 1011 The Algebra of Square Matrices750
 1012 Nonsingular Matrices756
 1013 Determinants760
 ‡1014 Proofs of Theorems on Determinants771
 ‡1015 Further Remarks on Determinants776
 †1016 The Method of Elimination781
 †1017 Matrices of Functions788
 †1018 Eigenvalues, Eigenvectors, Characteristic Polynomial of a Matrix790
 ‡1019 Matrix Representations of a Linear Mapping795
 ‡1020 Jordan Matrices796
 ‡1021 Similar Matrices799

CHAPTER 11 LINEAR EUCLIDEAN GEOMETRY
 Introduction804
 111 Inner Product and Norm in V₃805
 112 Unit Vectors, Angle Between Vectors807
 ‡113 Euclidean Vector Space of Dimension n808
 114 Points, Vectors, Distance, Lines in 3Dimensional Euclidean Space R³811
 ‡115 Lines in nDimensional Euclidean Space817
 116 The Cross Product (Vector Product)819
 117 Triple Products824
 118 Application of the Cross Product to Lines in Space826
 ‡119 The Cross Product in V_{n}828
 1110 Planes in R³831
 1111 Relations between Lines and Planes838
 1112 Relations between Two Planes840
 ‡1113 Hyperplanes and Linear Manifolds in Rⁿ842
 1114 Other Cartesian Coordinate Systems in R³844
 1115 Lengths, Areas, and Volumes in R³848
 ‡1116 New Coordinates and Volume in Rⁿ854
 1117 Linear Mappings of R³ into R³857
 ‡1118 Linear Mappings of Rⁿ into R^{m}860
 1119 Surfaces in R³862
 1120 Cylindrical and Spherical Coordinates865
 1121 Change of Coordinates in R³868
 ‡1122 Change of Coordinates in Rⁿ873

CHAPTER 12 DIFFERENTIAL CALCULUS OF FUNCTIONS OF SEVERAL VARIABLES
 Introduction875
 121 Sets in the Plane876
 122 Functions of Two Variables878
 123 Functions of Three or More Variables883
 124 Vector Functions884
 125 Matrix Functions886
 126 Operations on Functions887
 127 Limits and Continuity889
 128 Partial Derivatives897
 129 The Differential902
 1210 Chain Rules908
 1211 The Directional Derivative913
 1212 Differential of a Vector Function, the Jacobian Matrix918
 1213 The General Chain Rule922
 1214 Implicit Functions926
 ‡1215 Implicit Function Theorem935
 1216 Inverse Functions939
 1217 Curves in Space945
 1218 Surfaces in Space948
 1219 Partial Derivatives of Higher Order954
 ‡1220 Proof of Theorem on Mixed Partial Derivatives957
 1221 Taylor's Formula960
 1222 Maxima and Minima of Functions of Two Variables966
 ‡1223 Lagrange Multipliers974
 ‡1224 Proof of Theorem on Local Maxima and Minima976
 ‡1225 Some Deeper Results on Continuity980

CHAPTER 13 INTEGRAL CALCULUS OF FUNCTIONS OF SEVERAL VARIABLES
 131 The Double Integral988
 132 Theory of the Double Integral997
 ‡133 Proof that the Double Integral can be Represented as a Limit1005
 134 Double Integrals in Polar Coordinates1010
 ‡135 Other Curvilinear Coordinates1013
 136 Triple Integrals1017
 137 Triple Integrals in Cylindrical and Spherical Coordinates1023
 138 Further Properties of Multiple Integrals1031
 139 Surface Area1037
 1310 Other Applications of Multiple Integrals1041
 1311 Line Integrals1049
 1312 Green's Theorem1060
 1313 Curl and Divergence, Vector Form of Green's Theorem1062
 1314 Exact Differentials and Independence of Path1073
 1315 The Divergence Theorem and Stokes' Theorem in Space1081

CHAPTER 14 ORDINARY DIFFERENTIAL EQUATIONS
 141 Basic Concepts1088
 142 Graphical Method and Method of StepbyStep Integration1096
 143 Exact FirstOrder Equations1100
 144 Equations with Variables Separable and Equations of Form y' = g(y/x)403
 145 The Linear Equation of First Order1111
 146 Linear Differential Equations of Order n1117
 147 Variation of Parameters1124
 148 ComplexValued Solutions of Linear Differential Equations1126
 149 Homogeneous Linear Differential Equations with Constant Coefficients1129
 ‡1410 Linear Independence of Solutions of the Homogeneous Linear Equation with Constant Coefficients1132
 1411 Nonhomogeneous Linear Differential Equations with Constant Coefficients1134
 1412 Applications of Linear Differential Equations1137
 1413 Vibrations of a MassSpring System1142
 1414 Simultaneous Linear Differential Equations1148
 1415 Solutions Satisfying Initial Conditions, Variation of Parameters1154
 1416 ComplexValued Solutions of Systems of Linear Differential Equations1159
 1417 Homogeneous Linear Systems with Constant Coefficients1162
 1418 Nonhomogeneous Linear Systems with Constant Coefficients: Stability1166
 1419 Method of Elimination1171
 ‡1420 Application of Exponential Function of a Matrix1174
 1421 Autonomous Linear Systems of Order Two1179
 1422 Power Series Solutions1187
 1423 Numerical Solutions of Differential Equations1193
 ANSWERS TO SELECTED PROBLEMSA45
 INDEXI7