The second volume of this introduction into analysis deals with the integration theory of functions of one variable, the multidimensional differential calculus and the theory of curves and line integrals. The modern and clear development that started in Volume I is continued. In this way a sustainable basis is created which allows the reader to deal with interesting applications that sometimes go beyond material represented in traditional textbooks. This applies, for instance, to the exploration of Nemytskii operators which enable a transparent introduction into the calculus of variations and the derivation of the Euler-Lagrange equations.
InhaltsangabeIntegral calculus in one variable.- Multivariable differential calculus.- Line integrals.
Preface.- VI. Integral Calculus in One Variable - 1. Step Continuous Functions - 2. Continuous Extensions - 3. The Cauchy-Riemann Integral - 4. Properties of the Integral - 5. The Technology of Integration - 6. Sums and Integrals - 7. Fourier Series - 8. Improper Integrals - 9. The Gamma Function.- VII. Differential Calculus in Several Variables - 1. Continuous Linear Mappings - 2. Differentiability - 3. Calculation Rules - 4. Multilinear Mappings - 5. Higher Derivatives - 6. Nemytski Operators and Calculus of Variations - 7. Inverse Mappings - 8. Implicit Functions - 9. Manifolds - 10. Tangents and Normals.- VIII. Line Integrals - 1. Curves and Their Length - 2. Curves in Rn - 3. Pfaff Forms - 4. Line Integrals - 5. Holomorphic Functions - 6. Meromorphic Functions.- Bibliography.- Index.