| Preface | p. v |
| Differential Calculus | p. 1 |
| Differential Calculus of Scalar Functions | p. 1 |
| Directional and partial derivatives, and the differential | p. 1 |
| Directional derivatives | p. 1 |
| The differential | p. 2 |
| The gradient vector | p. 5 |
| Direction of steepest ascent | p. 6 |
| Directional derivatives and differential in coordinates | p. 7 |
| Partial derivatives | p. 7 |
| Jacobian matrix | p. 8 |
| The differential in the dual basis | p. 8 |
| The gradient vector in coordinates | p. 9 |
| The tangent plane | p. 9 |
| The orthogonal to the tangent space | p. 10 |
| The tangent map | p. 11 |
| Differentiability and blow-up | p. 12 |
| Differential Calculus for Vector-valued Functions | p. 12 |
| Differentiability | p. 14 |
| Jacobian matrix | p. 14 |
| The tangent space | p. 16 |
| The calculus | p. 20 |
| Differentiation of compositions | p. 21 |
| Calculus for matrix-valued maps | p. 22 |
| Theorems of Differential Calculus | p. 24 |
| Maps with continuous derivatives | p. 24 |
| Functions of class C1 (A) | p. 24 |
| Functions of class C1 (A) | p. 25 |
| Functions of class C2 (A) | p. 26 |
| Functions of classes Ck (A) and C&infinity; (A) | p. 28 |
| Mean value theorem | p. 29 |
| Scalar functions | p. 29 |
| Vector-valued functions | p. 31 |
| Taylor's formula | p. 32 |
| Taylor's formula of second order | p. 33 |
| Taylor formulas of higher order | p. 34 |
| Real analytic functions | p. 36 |
| A converse of Taylor's theorem | p. 37 |
| Critical points | p. 38 |
| Some classical partial differential equations | p. 42 |
| Invertibility of Maps Rn Rn | p. 46 |
| Banach's fixed point theorem | p. 47 |
| Local invertibility | p. 48 |
| A few examples | p. 51 |
| A variational proof of the inverse function theorem | p. 55 |
| Global invertibility | p. 56 |
| Differential Calculus in Banach Spaces | p. 57 |
| G&ahat;teaux and Fréchet differentials | p. 57 |
| Gradient | p. 59 |
| Mean value theorem | p. 60 |
| Higher order derivatives and Taylor's formula | p. 61 |
| Local invertibility in Banach spaces | p. 62 |
| Exercises | p. 62 |
| Integral Calculus | p. 67 |
| Lebesgue's Integral | p. 67 |
| Definitions and properties: a short summary | p. 67 |
| Lebesgue's measure | p. 68 |
| Measurable functions | p. 70 |
| Lebesgue's integral | p. 71 |
| Basic properties of Lebesgue's Integral | p. 72 |
| The integral as area of the subgraph | p. 74 |
| Chebyshev's inequality | p. 74 |
| Negligible sets and the integral | p. 74 |
| Riemann integrable functions | p. 75 |
| Fubini's theorem and reduction to iterated integrals | p. 76 |
| Change of variables | p. 78 |
| Differentiation and primitives | p. 78 |
| Convergence Theorems | p. 81 |
| Monotone convergence | p. 81 |
| Dominated convergence | p. 83 |
| Absolute continuity of the integral | p. 86 |
| Differentiation under the integral sign | p. 86 |
| Mollifiers and Approximations | p. 89 |
| C0-approximations and Lusin's theorem | p. 89 |
| Mollifying in Rn | p. 91 |
| Mollifying in ¿ | p. 94 |
| Calculus of Integrals | p. 96 |
| Calculus of multiple integrals | p. 96 |
| Normal sets | p. 97 |
| Rotational figures | p. 99 |
| Changes of coordinates | p. 100 |
| Measure of the n-dimensional ball | p. 104 |
| Isodiametric inequality | p. 105 |
| Euler's ¿ function | p. 106 |
| Tetrahedrons | p. 108 |
| Monte Carlo method | p. 110 |
| Differentiation under the integral sign | p. 111 |
| Measure and Area | p. 114 |
| Hausdorff's measures | p. 114 |
| Area formula | p. 116 |
| Calculus of the area of a surface | p. 119 |
| The coarea formula | p. 121 |
| Gauss-Green Formulas | p. 123 |
| Two simple situations | p. 124 |
| Admissible sets | p. 126 |
| Decomposition of unity | p. 127 |
| Gauss-Green formulas | p. 128 |
| Integration by parts | p. 130 |
| The divergence theorem | p. 130 |
| Geometrical meaning of the divergence | p. 130 |
| Divergence and transport of volume | p. 131 |
| Exercises | p. 132 |
| Curves and Differential Forms | p. 137 |
| Differential Forms, Vector Fields, and Work | p. 137 |
| Vector fields and differential forms | p. 137 |
| Curves | p. 138 |
| Integration along a curve and work | p. 140 |
| Conservative Fields and Potentials | p. 142 |
| Exact differential forms | p. 142 |
| Closed Forms and Irrotational Fields | p. 145 |
| Closed forms | p. 145 |
| Poincaré lemma | p. 147 |
| Homotopic curves and work | p. 148 |
| Simply connected subsets and closed forms | p. 150 |
| Pull back of a differential form | p. 151 |
| Homotopy formula | p. 153 |
| Stokes's theorem in a square | p. 153 |
| Homotopy formula | p. 155 |
| Stokes's Formula in the Plane | p. 156 |
| Exercises | p. 158 |
| Holomorphic Functions | p. 159 |
| Functions from C to C | p. 159 |
| Complex numbers | p. 159 |
| Complex derivative | p. 159 |
| Cauchy-Riemann equations | p. 160 |
| The Fundamental Theorem of Calculus on C | p. 163 |
| Line integrals | p. 163 |
| Holomorphic primitives and line integrals | p. 164 |
| Fundamental Theorems about Holomorphic Functions | p. 167 |
| Goursat and Cauchy theorems | p. 167 |
| Goursat lemma | p. 167 |
| Elementary domains and Goursat's theorem | p. 168 |
| Cauchy formula and power series development | p. 171 |
| Liouville's theorem | p. 175 |
| The unique continuation principle | p. 176 |
| Holomorphic differentials | p. 176 |
| Winding number | p. 178 |
| Stokes's formula and Cauchy's and Morera's theorems | p. 179 |
| Examples of Holomorphic Functions | p. 180 |
| Some simple functions | p. 180 |
| Inverses of holomorphic functions | p. 182 |
| Complex logarithm | p. 183 |
| Real powers | p. 185 |
| Singularities | p. 187 |
| Removable singularities | p. 188 |
| Poles | p. 189 |
| Essential singularities | p. 190 |
| Singularities at infinity | p. 191 |
| Singular points at boundary and radius of convergence | p. 192 |
| Laurent series development | p. 192 |
| Residues | p. 195 |
| Calculus of residues | p. 196 |
| Definite integrals by the residue method | p. 197 |
| Sums of series by the residue method | p. 204 |
| Z-transform | p. 208 |
| Z-transform of a sequence of vectors | p. 213 |
| Systems of recurrences and Z-transform | p. 215 |
| Further Consequences of Cauchy's Formula | p. 217 |
| The argument principle | p. 217 |
| Rouché's theorem | p. 218 |
| Maximum principle | p. 219 |
| On the convergence of holomorphic functions | p. 220 |
| Schwarz's lemma | p. 221 |
| Open mapping and the inverse theorem | p. 221 |
| Biholomorphisms | p. 222 |
| Riemann mapping theorem | p. 223 |
| Harmonic functions and Riemann's mapping theorem | p. 225 |
| Schwarz's and Poisson's formulas | p. 226 |
| Hilbert's transform | p. 227 |
| Exercises | p. 229 |
| Surfaces and Level Sets | p. 237 |
| Immersed and Embedded Surfaces | p. 237 |
| Diffeomorphisms | p. 237 |
| Tangent vectors | p. 239 |
| r-dimensional surfaces in Rn | p. 241 |
| Submanifolds | p. 241 |
| Immersions | p. 241 |
| Parameterizations of maximal rank | p. 243 |
| Implicit Function Theorem | p. 247 |
| Implicit functions | p. 247 |
| The theorem | p. 249 |
| Foliations | p. 253 |
| Submersions | p. 257 |
| Irregular level sets | p. 258 |
| Some Applications | p. 259 |
| Small perturbations | p. 259 |
| Quadratic systems | p. 260 |
| Nonlinear Cauchy problem | p. 260 |
| A boundary value problem | p. 261 |
| C1-dependence on initial data | p. 261 |
| Rectifiability theorem for vector fields | p. 262 |
| Critical points and critical values: Sard's lemma | p. 264 |
| Morse lemma | p. 266 |
| Gradient flow | p. 267 |
| Constrained critical points: the multiplier rule | p. 268 |
| Some applications | p. 270 |
| Orthogonal projection and eigenvectors | p. 270 |
| Inequalities | p. 272 |
| Lyapunov-Schmidt procedure | p. 275 |
| Maps with locally constant rank and functional dependence | p. 279 |
| Curvature of Curves and Surfaces | p. 281 |
| Curvature of a curve in Rn | p. 281 |
| Moving frame for a planar curve | p. 284 |
| Moving frame of a curve in R3 | p. 286 |
| Curvature of a submanifold of Rn | p. 288 |
| First fundamental form | p. 289 |
| Second fundamental form | p. 290 |
| Curvature vector | p. 291 |
| Mean curvature vector | p. 292 |
| Curvature of surfaces of codimension one | p. 293 |
| Gradient and divergence on a surface | p. 296 |
| First variation of the area | p. 299 |
| Laplace-Beltrami operator and the mean curvature | p. 300 |
| Distance function | p. 301 |
| Exercises | p. 303 |
| Systems of Ordinary Differential Equations | p. 309 |
| Linear Systems | p. 310 |
| Linear systems of first-order ODEs | p. 310 |
| Linear systems with constant coefficients | p. 313 |
| More about linear systems | p. 314 |
| Higher-order equations | p. 318 |
| Higher-order equations and first-order systems | p. 318 |
| Homogeneous linear equations with constant coefficients | p. 319 |
| Nonhomogeneous linear ODEs | p. 321 |
| Stability | p. 322 |
| Critical points and linearization | p. 322 |
| Lyapunov's method | p. 325 |
| Poincaré-Bendixson Theorem | p. 328 |
| Limit sets and invariant sets | p. 329 |
| Poincaré-Bendixson theorem | p. 331 |
| Systems on a torus | p. 332 |
| Exercises | p. 336 |
| Mathematicians and Other Scientists | p. 339 |
| Bibliographical Notes | p. 341 |
| Index | p. 343 |
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