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# Calculus of several variables by Serge Lang 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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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  for the proof. 3. Lipschitz estimates In the paper  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.  V. Maz’ya, Sobolev spaces, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985.  V. Maz’ya, Classes of regions and embedding theorems for functional spaces, Dokl. Akad. Nauk SSSR 133 (1960), 527–530.  F. , San Diego, CA, 2000. Yuri Burago Saint-Petersburg branch of Steklov Math Institute Fontanka 27 St. ru Nikolay N.

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