10.12 Comparison tests for series of nonnegative terms
10.13 The integral test
10.14 Exercises
10.15 The root test and the ratio test for series of nonnegative terms
10.16 Exercises
10.17 Alternating series
10.18 Conditional and absolute convergence
10.19 The convergence tests of Dirichlet and Abel
10.20 Exercises
*10.21 Rearrangements of series
10.22 Miscellaneous review exercises
10.23 Improper integrals
10.24 Exercises
11. SEQUENCES AND SERIES OF FUNCTIONS
11.1 Pointwise convergence of sequences of functions
11.2 Uniform convergence of sequences of functions
11.3 Uniform convergence and continuity
11.4 Uniform convergence and integration
11.5 A sufficient condition for uniform convergence
11.6 Power series. Circle of convergence
11.7 Exercises
11.8 Properties of functions represented by real power series
11.9 The Taylor's series generated by a function
11.10 A sufficient condition for convergence of a Taylor's series
11.11 Power-series expansions for the exponential and trigonometric functions
*11.12 Bernstein's theorem
11.13 Exercises
11.14 Power series and differential equations
11.15 The binomial series
11.16 Exercises
12. VECTOR ALGEBRA
12.1 Historical introduction
12.2 The vector space of n-tuples of real numbers
12.3 Geometric interpretation for n ≤ 3
12.4 Exercises
12.5 The dot product
12.6 Length or norm of a vector
12.7 Orthogonality of vectors
12.8 Exercises
12.9 Projections. Angle between vectors in n-space
12.10 The unit coordinate vectors
12.11 Exercises
12.12 The linear span of a finite set of vectors
12.13 Linear independence
12.14 Bases
12.15 Exercises
12.16 The vector space Vn(C) of n-tuples of complex numbers
12.17 Exercises
13. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY
13.1 Introduction
13.2 Lines in n-space
13.3 Some simple properties of straight lines
13.4 Lines and vector-valued functions
13.5 Exercises
13.6 Planes in Euclidean n-space
13.7 Planes and vector-valued functions
13.8 Exercises
13.9 The cross product
13.10 The cross product expressed as a determinant
13.11 Exercises
13.12 The scalar triple product
13.13 Cramer's rule for solving a system of three linear equations
13.14 Exercises
13.15 Normal vectors to planes
13.16 Linear Cartesian equations for planes
13.17 Exercises
13.18 The conic sections
13.19 Eccentricity of conic sections
13.20 Polar equations for conic sections
13.21 Exercises
13.22 Conic sections symmetric about the origin
13.23 Cartesian equations for the conic sections
13.24 Exercises
13.25 Miscellaneous exercises on conic sections
14. CALCULUS OF VECTOR-VALUED FUNCTIONS
14.1 Vector-valued functions of a real variable
14.2 Algebraic operations. Components
14.3 Limits, derivatives, and integrals
14.4 Exercises
14.5 Applications to curves. Tangency
14.6 Applications to curvilinear motion. Velocity, speed, and acceleration
14.7 Exercises
14.8 The unit tangent, the principal normal, and the osculating plane of a curve
14.9 Exercises
14.10 The definition of arc length
14.11 Additivity of arc length
14.12 The arc-length function
14.13 Exercises
14.14 Curvature of a curve
14.15 Exercises
14.16 Velocity and acceleration in polar coordinates
14.17 Plane motion with radial acceleration
14.18 Cylindrical coordinates
14.19 Exercises
14.20 Applications to planetary motion
14.21 Miscellaneous review exercises
15. LINEAR SPACES
15.1 Introduction
15.2 The definition of a linear space
15.3 Examples of linear spaces
15.4 Elementary consequences of the axioms
15.5 Exercises
15.6 Subspaces of a linear space
15.7 Dependent and independent sets in a linear space
15.8 Bases and dimension
15.9 Exercises
15.10 Inner products, Euclidean spaces, norms
15.11 Orthogonality in a Euclidean space
15.12 Exercises
15.13 Construction of orthogonal sets. The Gram–Schmidt process
15.14 Orthogonal complements. Projections
15.15 Best approximation of elements in a Euclidean space by elements in a finite-dimensional subspace
15.16 Exercises
16. LINEAR TRANSFORMATIONS AND MATRICES
16.1 Linear transformations
16.2 Null space and range
16.3 Nullity and rank
16.4 Exercises
16.5 Algebraic operations on linear transformations
16.6 Inverses
16.7 One-to-one linear transformations
16.8 Exercises
16.9 Linear transformations with prescribed values
16.10 Matrix representations of linear transformations
16.11 Construction of a matrix representation in diagonal form
16.12 Exercises
16.13 Linear spaces of matrices
16.14 Isomorphism between linear transformations and matrices
16.15 Multiplication of matrices
16.16 Exercises
16.17 Systems of linear equations
16.18 Computation techniques
16.19 Inverses of square matrices
16.20 Exercises
16.21 Miscellaneous exercises on matrices
ANSWERS TO EXERCISES
INDEX
END USER LICENSE AGREEMENT
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