[...]... Nonhomogeneous linear systems with constant coefficients 7.17 7.18 7.19 7 .20 7 .21 7 .22 7 .23 7 .24 21 3 21 5 Exercises 21 7 The general linear system Y’(t) = P(t) Y(t) + Q(t) 22 0 A power-series method for solving homogeneous linear systems 22 1 Exercises 22 2 Proof of the existence theorem by the method of successive approximations The method of successive approximations applied to first-order nonlinear systems... systems 22 7 22 9 Proof of an existence-uniqueness theorem for first-order nonlinear systems 23 0 Exercises 23 2 Successive approximations and fixed points of operators 23 3 Normed linear spaces *7 .25 *7 .26 k7 .27 Contraction operators k7 .28 Fixed-point theorem for contraction operators A7 .29 Applications of the fixed-point theorem 23 4 23 5 23 7 PART 2 NONLINEAR ANALYSIS 8 DIFFERENTIAL CALCULUS OF SCALAR AND VECTOR... equality of mixed partial derivatives Miscellaneous exercises 25 1 25 8 25 9 26 1 26 2 26 3 26 6 26 8 26 9 27 1 27 2 27 3 27 5 27 7 28 1 9 APPLICATIONS OF THE DIFFERENTIAL CALCULUS 9.1 9 .2 Partial differential equations A first-order partial differential equation with constant coefficients 9.3 Exercises 9.4 The one-dimensional wave equation 9.5 Exercises 9.6 Derivatives of functions defined implicitly 9.7 Worked examples... 12. 10 12. 11 12. 12 12. 13 12. 14 12. 15 * 12. 16 * 12. 17 12. 18 12. 19 12. 20 12. 21 Exercises The theorem of Stokes The curl and divergence of a vector field Exercises Further properties of the curl and divergence Exercises Reconstruction of a vector field from its curl Exercises Extensions of Stokes’ theorem The divergence theorem (Gauss’ theorem:) Applications of the divergence theorem Exercises 417 420 423 ... a real quadratic form to a diagonal form 5.14 Applications to analytic geometry 5.15 Exercises A5.16 Eigenvalues of a symmetric transformation obtained as values of its quadratic form k5.17 Extremal properties of eigenvalues of a symmetric transformation k5.18 The finite-dimensional case 5.19 Unitary transformations 5 .20 Exercises 114 115 117 117 118 120 121 122 122 123 124 126 128 130 134 135 136 137... examples 9.8 Exercises 9.9 Maxima, minima, and saddle points 9 IO Second-order Taylor formula for scalar fields 28 3 28 4 28 6 28 8 29 2 29 4 29 8 3 02 303 308 9.11 The nature of a stationary point determined by the eigenvalues of the Hessian 310 matrix 3 12 9. 12 Second-derivative test for extrema of functions of two variables 313 9.13 Exercises 314 9.14 Extrema with constraints Lagrange’s multipliers 318 9 I5 Exercises... set of all real numbers, and let x + y and ax be ordinary addition and multiplication of real numbers EXAMPLE 2 Let V = C, the set of all complex numbers, define x + y to be ordinary addition of complex numbers, and define ax to be multiplication of the complex number x Examples of linear spaces by the real number a Even though the elements of V are complex numbers, this is a real linear space because... Distributions of two-dimensional random variables 14 .20 Two-dimensional discrete distributions 14 .21 Two-dimensional continuous distributions Density functions 14 .22 Exercises 14 .23 Distributions of functions of two random variables 14 .24 Exercises 14 .25 Expectation and variance 14 .26 Expectation of a function of a random variable 14 .27 Exercises 14 .28 Chebyshev’s inequality 14 .29 Laws of large numbers 14.30... 15.19 15 .20 15 .21 15 .22 15 .23 Chebyshev polynomials A minimal property of Chebyshev polynomials Application to the error formula for interpolation Exercises Approximate integration The trapezoidal rule Simpson’s rule Exercises The Euler summation formula Exercises Suggested References Answers to exercises Index xxi 596 598 599 600 6 02 605 610 613 618 621 622 665 Calculus PART 1 LINEAR ANALYSIS 1 LINEAR. .. called a complex linear space Sometimes a linear space is referred to as a linear vector space or simply a vector space; the numbers used as multipliers are also called scalars A real linear space has real numbers as scalars; a complex linear space has complex numbers as scalars Although we shall deal primarily with examples of real linear spaces, all the theorems are valid for complex linear spaces . fields 8 .21 Matrix form of the chain rule 8 .22 Exercises A8 .23 Sufficient conditions for the equality of mixed partial derivatives 8 .24 Miscellaneous exercises 25 1 25 8 25 9 26 1 26 2 26 3 26 6 26 8 26 9 27 1 27 2 27 3 27 5 27 7 28 1 9 for solving homogeneous linear systems 22 0 7 .20 Exercises 22 1 7 .21 Proof of the existence theorem by the method of successive approximations 22 2 7 .22 The method of successive approximations applied. subspace 28 1.17 Exercises 30 2. LINEAR TRANSFORMATIONS AND MATRICES 2. 1 Linear transformations 2. 2 Null space and range 2. 3 Nullity and rank 31 32 34 xi xii Contents 2. 4 Exercises 2. 5 Algebraic