# Mathematical Analysis An Introduction to Functions of Several Variables by Mariano Giaquinta

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## Mathematical Analysis An Introduction to Functions of Several Variables by Mariano Giaquinta and Giuseppe Modica pdf free download

Mathematical Analysis An Introduction to Functions of Several Variables by Mariano Giaquinta and Giuseppe Modica pdf free download. This book introduces the main ideas and fundamental methods of differential and integral calculus for functions of several variables. In Chapter 1 we discuss differential calculus for functions of several variables with a short excursion into differential calculus in Banach spaces. In Chapter 2 we present some of the most relevant results of the Lebesgue integration theory, including the limit and approximation theorems, Fubini’s theorem, the area and coarea theorems, and Gauss–Green formulas.

### Mathematical Analysis An Introduction to Functions of Several Variables by Mariano Giaquinta and Giuseppe Modica pdf download

The aim is to provide the reader with all that is needed to use the power of Lebesgue integration. For this reason some details as well as some proofs concerning the formulation of the theory are skipped, as we think they are more appropriate in the general context of measure theory. In Chapter 3 we deal with potentials and integration of differential 1-forms, focusing on solenoidal and irrotational fields. Chapter 4 provides a sufficiently wide introduction to the theory of holomorphic functions of one complex variable. We present the fundamental theorems and discuss singularities and residues as well as Riemann’s theorem on conformal representation and the related Schwarz and Poisson formulas and Hilbert’s transform.

#### Mathematical Analysis An Introduction to Functions of Several Variables by Mariano Giaquinta and Giuseppe Modica pdf download

In Chapter 5, we discuss the notions of immersed and embedded surface in Rn, and we present the implicit function theorem and some of its applications to vector fields, constrained minimization, and functional dependence. The chapter ends with the study of some notions of the local theory of curves and surfaces, such as of curvature, first variation of area, the Laplace–Beltrami operator, and distance function. In Chapter 6, after a few preliminaries about systems of linear ordinary differential equations, we discuss a few results concerning the stability of nonlinear systems and the Poincar´e–Bendixson theorem in order to show that dynamical systems with one degree of freedom do not present chaos, in contrast with the one-dimensional discrete dynamics or the higher-dimensional continuous dynamics.

## Mathematical Analysis An Introduction to Functions of Several Variables by Mariano Giaquinta and Giuseppe Modica pdf download 