This book is dedicated to Sergei Mikhailovich Nikol'skii on the occasion of his eighty-fifth birthday. The collection contains new results on the following topics: approximation of functions, imbedding theory, interpolation of function spaces, convergence of series in trigonometric and general orthogonal systems, quasilinear elliptic problems, spectral theory of nonselfadjoint operators, asymptotic properties of pseudodifferential operators, and methods of approximate solution of Laplace's equation.

This text was produced for the second part of a two-part sequence on advanced calculus, whose aim is to provide a firm logical foundation for analysis. The first part treats analysis in one variable, and the text at hand treats analysis in several variables. After a review of topics from one-variable analysis and linear algebra, the text treats in succession multivariable differential calculus, including systems of differential equations, and multivariable integral calculus. It builds on this to develop calculus on surfaces in Euclidean space and also on manifolds. It introduces differential forms and establishes a general Stokes formula. It describes various applications of Stokes formula, from harmonic functions to degree theory. The text then studies the differential geometry of surfaces, including geodesics and curvature, and makes contact with degree theory, via the Gauss–Bonnet theorem. The text also takes up Fourier analysis, and bridges this with results on surfaces, via Fourier analysis on spheres and on compact matrix groups.