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| Foreword | p. v |
| Integral calculus in one variable | |
| Jump continuous functions | p. 4 |
| Staircase and jump continuous functions | p. 4 |
| A characterization of jump continuous functions | p. 6 |
| The Banach space of jump continuous functions | p. 7 |
| Continuous extensions | p. 10 |
| The extension of uniformly continuous functions | p. 10 |
| Bounded linear operators | p. 12 |
| The continuous extension of bounded linear operators | p. 15 |
| The Cauchy-Riemann Integral | p. 17 |
| The integral of staircase functions | p. 17 |
| The integral of jump continuous functions | p. 19 |
| Riemann sums | p. 20 |
| Properties of integrals | p. 25 |
| Integration of sequences of functions | p. 25 |
| The oriented integral | p. 26 |
| Positivity and monotony of integrals | p. 27 |
| Componentwise integration | p. 30 |
| The first fundamental theorem of calculus | p. 30 |
| The indefinite integral | p. 32 |
| The mean value theorem for integrals | p. 33 |
| The technique of integration | p. 38 |
| Variable substitution | p. 38 |
| Integration by parts | p. 40 |
| The integrals of rational functions | p. 43 |
| Sums and integrals | p. 50 |
| The Bernoulli numbers | p. 50 |
| Recursion formulas | p. 52 |
| The Bernoulli polynomials | p. 53 |
| The Euler-Maclaurin sum formula | p. 54 |
| Power sums | p. 56 |
| Asymptotic equivalence | p. 57 |
| The Riemann [zeta] function | p. 59 |
| The trapezoid rule | p. 64 |
| Fourier series | p. 67 |
| The L[subscript 2] scalar product | p. 67 |
| Approximating in the quadratic mean | p. 69 |
| Orthonormal systems | p. 71 |
| Integrating periodic functions | p. 72 |
| Fourier coefficients | p. 73 |
| Classical Fourier series | p. 74 |
| Bessel's inequality | p. 77 |
| Complete orthonormal systems | p. 79 |
| Piecewise continuously differentiable functions | p. 82 |
| Uniform convergence | p. 83 |
| Improper integrals | p. 90 |
| Admissible functions | p. 90 |
| Improper integrals | p. 90 |
| The integral comparison test for series | p. 93 |
| Absolutely convergent integrals | p. 94 |
| The majorant criterion | p. 95 |
| The gamma function | p. 98 |
| Euler's integral representation | p. 98 |
| The gamma function on C(-N) | p. 99 |
| Gauss's representation formula | p. 100 |
| The reflection formula | p. 104 |
| The logarithmic convexity of the gamma function | p. 105 |
| Stirling's formula | p. 108 |
| The Euler beta integral | p. 110 |
| Multivariable differential calculus | |
| Continuous linear maps | p. 118 |
| The completeness of L (E, F) | p. 118 |
| Finite-dimensional Banach spaces | p. 119 |
| Matrix representations | p. 122 |
| The exponential map | p. 125 |
| Linear differential equations | p. 128 |
| Gronwall's lemma | p. 129 |
| The variation of constants formula | p. 131 |
| Determinants and eigenvalues | p. 133 |
| Fundamental matrices | p. 136 |
| Second order linear differential equations | p. 140 |
| Differentiability | p. 149 |
| The definition | p. 149 |
| The derivative | p. 150 |
| Directional derivatives | p. 152 |
| Partial derivatives | p. 153 |
| The Jacobi matrix | p. 155 |
| A differentiability criterion | p. 156 |
| The Riesz representation theorem | p. 158 |
| The gradient | p. 159 |
| Complex differentiability | p. 162 |
| Multivariable differentiation rules | p. 166 |
| Linearity | p. 166 |
| The chain rule | p. 166 |
| The product rule | p. 169 |
| The mean value theorem | p. 169 |
| The differentiability of limits of sequences of functions | p. 171 |
| Necessary condition for local extrema | p. 171 |
| Multilinear maps | p. 173 |
| Continuous multilinear maps | p. 173 |
| The canonical isomorphism | p. 175 |
| Symmetric multilinear maps | p. 176 |
| The derivative of multilinear maps | p. 177 |
| Higher derivatives | p. 180 |
| Definitions | p. 180 |
| Higher order partial derivatives | p. 183 |
| The chain rule | p. 185 |
| Taylor's formula | p. 185 |
| Functions of m variables | p. 186 |
| Sufficient criterion for local extrema | p. 188 |
| Nemytskii operators and the calculus of variations | p. 195 |
| Nemytskii operators | p. 195 |
| The continuity of Nemytskii operators | p. 195 |
| The differentiability of Nemytskii operators | p. 197 |
| The differentiability of parameter-dependent integrals | p. 200 |
| Variational problems | p. 202 |
| The Euler-Lagrange equation | p. 204 |
| Classical mechanics | p. 207 |
| Inverse maps | p. 212 |
| The derivative of the inverse of linear maps | p. 212 |
| The inverse function theorem | p. 214 |
| Diffeomorphisms | p. 217 |
| The solvability of nonlinear systems of equations | p. 218 |
| Implicit functions | p. 221 |
| Differentiable maps on product spaces | p. 221 |
| The implicit function theorem | p. 223 |
| Regular values | p. 226 |
| Ordinary differential equations | p. 226 |
| Separation of variables | p. 229 |
| Lipschitz continuity and uniqueness | p. 233 |
| The Picard-Lindelof theorem | p. 235 |
| Manifolds | p. 242 |
| Submanifolds of R[superscript n] | p. 242 |
| Graphs | p. 243 |
| The regular value theorem | p. 243 |
| The immersion theorem | p. 244 |
| Embeddings | p. 247 |
| Local charts and parametrizations | p. 252 |
| Change of charts | p. 255 |
| Tangents and normals | p. 260 |
| The tangential in R[superscript n] | p. 260 |
| The tangential space | p. 261 |
| Characterization of the tangential space | p. 265 |
| Differentiable maps | p. 266 |
| The differential and the gradient | p. 269 |
| Normals | p. 271 |
| Constrained extrema | p. 272 |
| Applications of Lagrange multipliers | p. 273 |
| Line integrals | |
| Curves and their lengths | p. 281 |
| The total variation | p. 281 |
| Rectifiable paths | p. 282 |
| Differentiable curves | p. 284 |
| Rectifiable curves | p. 286 |
| Curves in R[superscript n] | p. 292 |
| Unit tangent vectors | p. 292 |
| Parametrization by arc length | p. 293 |
| Oriented bases | p. 294 |
| The Frenet n-frame | p. 295 |
| Curvature of plane curves | p. 298 |
| Identifying lines and circles | p. 300 |
| Instantaneous circles along curves | p. 300 |
| The vector product | p. 302 |
| The curvature and torsion of space curves | p. 303 |
| Pfaff forms | p. 308 |
| Vector fields and Pfaff forms | p. 308 |
| The canonical basis | p. 310 |
| Exact forms and gradient fields | p. 312 |
| The Poincare lemma | p. 314 |
| Dual operators | p. 316 |
| Transformation rules | p. 317 |
| Modules | p. 321 |
| Line integrals | p. 326 |
| The definition | p. 326 |
| Elementary properties | p. 328 |
| The fundamental theorem of line integrals | p. 330 |
| Simply connected sets | p. 332 |
| The homotopy invariance of line integrals | p. 333 |
| Holomorphic functions | p. 339 |
| Complex line integrals | p. 339 |
| Holomorphism | p. 342 |
| The Cauchy integral theorem | p. 343 |
| The orientation of circles | p. 344 |
| The Cauchy integral formula | p. 345 |
| Analytic functions | p. 346 |
| Liouville's theorem | p. 348 |
| The Fresnel integral | p. 349 |
| The maximum principle | p. 350 |
| Harmonic functions | p. 351 |
| Goursat's theorem | p. 353 |
| The Weierstrass convergence theorem | p. 356 |
| Meromorphic functions | p. 360 |
| The Laurent expansion | p. 360 |
| Removable singularities | p. 364 |
| Isolated singularities | p. 365 |
| Simple poles | p. 368 |
| The winding number | p. 370 |
| The continuity of the winding number | p. 374 |
| The generalized Cauchy integral theorem | p. 376 |
| The residue theorem | p. 378 |
| Fourier integrals | p. 379 |
| References | p. 387 |
| Index | p. 389 |
| Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9783764374723
ISBN-10: 3764374721
Published: 16th May 2008
Format: Paperback
Language: English
Number of Pages: 416
Audience: Professional and Scholarly
Publisher: Springer Nature B.V.
Country of Publication: GB
Dimensions (cm): 24.13 x 17.15 x 1.91
Weight (kg): 0.79
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