By Serge Lang

It is a new, revised variation of this widely recognized textual content. the entire simple issues in calculus of a number of variables are lined, together with vectors, curves, services of numerous variables, gradient, tangent aircraft, maxima and minima, strength services, curve integrals, Green's theorem, a number of integrals, floor integrals, Stokes' theorem, and the inverse mapping theorem and its results. The presentation is self-contained, assuming just a wisdom of simple calculus in a single variable. Many thoroughly worked-out difficulties were incorporated.

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**Example text**

The H¨olderianity up to ∂Ω can be obtained under smoothness assumptions on the boundary of Ω . 7. If u : B → R is a continuous function such that |x − y| |u(x) − u(y)| ≤ K + |x − y|α (d∂B (x) ∧ d∂B (y))1−α for some K > 0 and 0 < α < 1, then u can be extended up to B as a function satisfying |u(x) − u(y)| ≤ K |x − y|α for all x, y ∈ B where K ≥ K is a constant depending on α, K. We refer to [7] for the proof. 3. Lipschitz estimates In the paper [12] the authors consider C 2 solutions of the model equation −Δu + |Du|p + λu = f (x) , x ∈ Ω .

If moreover A is a real coeﬃcient operator, L is Lp -dissipative for any p.

Federer, Geometric measure theory, Springer, Berlin, 1969. H. W. Rishel, An integral formula for total gradient variation, Arch. Math. 11 (1960), 218–222. [5] V. Maz’ya, Sobolev spaces, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985. [6] V. Maz’ya, Classes of regions and embedding theorems for functional spaces, Dokl. Akad. Nauk SSSR 133 (1960), 527–530. [7] F. , San Diego, CA, 2000. Yuri Burago Saint-Petersburg branch of Steklov Math Institute Fontanka 27 St. ru Nikolay N.