The moment quantity of this advent into research bargains with the mixing concept of services of 1 variable, the multidimensional differential calculus and the idea of curves and line integrals. the fashionable and transparent improvement that began in quantity I is sustained. during this approach a sustainable foundation is created which permits the reader to house attention-grabbing functions that typically transcend fabric represented in conventional textbooks. this is applicable, for example, to the exploration of Nemytskii operators which allow a clear creation into the calculus of diversifications and the derivation of the Euler-Lagrange equations.
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Additional info for Analysis II
Viii Contents Sums and integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 The Bernoulli numbers . . . . . . . Recursion formulation . . . . . . . . . The Bernoulli polynomials . . . . . The Euler–Maclaurin sum formulation energy sums . . . . . . . . . . . . . Asymptotic equivalence . . . . . . . The Riemann ζ functionality . . . . . . The trapezoid rule . . . . . . . . . . . . . . . . . 50 fifty two fifty three fifty four fifty six fifty seven fifty nine sixty four Fourier sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sixty seven The L2 scalar product . . . . . . . . . . . . . Approximating within the quadratic suggest . . . . . Orthonormal structures . . . . . . . . . . . . . . Integrating periodic features . . . . . . . . . Fourier coefficients . . . . . . . . . . . . . . . Classical Fourier sequence . . . . . . . . . . . . . Bessel’s inequality . . . . . . . . . . . . . . . . entire orthonormal structures . . . . . . . . Piecewise constantly differentiable services Uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sixty seven sixty nine seventy one seventy two seventy three seventy four seventy seven seventy nine eighty two eighty three eight incorrect integrals . . . . . . . Admissible capabilities . . . . . . incorrect integrals . . . . . . . . The necessary comparability attempt for totally convergent integrals The majorant criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ninety ninety ninety ninety three ninety four ninety five nine The gamma functionality . . . . . . . . . . . . . . . . Euler’s vital illustration . . . . . . . . . . . The gamma functionality on C\(−N) . . . . . . . . . Gauss’s illustration formulation . . . . . . . . . . . The reflection formulation . . . . . . . . . . . . . . . The logarithmic convexity of the gamma functionality Stirling’s formulation . . . . . . . . . . . . . . . . . . The Euler beta critical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ninety eight ninety eight ninety nine a hundred 104 one hundred and five 108 a hundred and ten 6 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents ix bankruptcy VII Multivariable differential calculus 1 non-stop linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 The completeness of L(E, F ) . . . . . . . Finite-dimensional Banach areas . . . . Matrix representations . . . . . . . . . . The exponential map . . . . . . . . . . . Linear differential equations . . . . . . . Gronwall’s lemma . . . . . . . . . . . . . the difference of constants formulation . . . Determinants and eigenvalues . . . . . . primary matrices . . . . . . . . . . moment order linear differential equations 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 119 122 a hundred twenty five 128 129 131 133 136 a hundred and forty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 one hundred fifty 152 153 one hundred fifty five 156 158 159 162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 166 169 169 171 171 Multilinear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 non-stop multilinear maps . . . . The canonical isomorphism . . . . . Symmetric multilinear maps .