By Serge Lang

It is a new, revised variation of this widely recognized textual content. all the simple themes in calculus of a number of variables are coated, together with vectors, curves, capabilities 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 effects. The presentation is self-contained, assuming just a wisdom of uncomplicated calculus in a single variable. Many thoroughly worked-out difficulties were integrated.

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**Sample text**

2) ω+γr,η where r > 0, η ∈ ]π/2, θ[, and γr,η is the curve {λ ∈ C : |argλ| = η, |λ| ≥ r} ∪ {λ ∈ C : |argλ| ≤ η, |λ| = r}, oriented counterclockwise. We also set e0A x = x, ∀x ∈ X. A. 3) 33 34 Chapter 2. Analytic semigroups and intermediate spaces Since the function λ → etλ R(λ, A) is holomorphic in Sθ,ω , the deﬁnition of etA is independent of the choice of r and η. 1 that the mapping t → etA is analytic from ]0, +∞[ to L(X), and moreover it enjoys the semigroup property etA esA = e(t+s)A , ∀ t, s ≥ 0.

3, for every r > 0 the resolvent set of A contains the open ball centered at ω + ir with radius |ω + ir|/M . The union of such balls contains the sector S = {λ = ω : |arg(λ − ω)| < π − arctan M }. 4) gives ∞ R(λ, A) ≤ |λ − (ω + ir)|n n=0 M n+1 2M . ≤ r (ω 2 + r2 )(n+1)/2 On the other hand, for λ = ω + ir − θr/M it holds r ≥ (1/(4M 2 ) + 1)−1/2 |λ − ω|, so that R(λ, A) ≤ 2M (1/(4M 2 ) + 1)−1/2 |λ − ω|−1 . The statement follows. 1 and the Reiteration Theorem imply that for all 0 < θ < 1 and 1 ≤ p ≤ ∞ such that kθ/n is not integer we have (X, D(Ak ))θ,p = (X, D(An ))kθ/n,p , (X, D(Ak ))θ = (X, D(An ))kθ/n .

2)(i) we get (X, Y )1 = (X, Y )1,p = {0}, p < ∞. Therefore, from now on we shall consider the cases (θ, p) ∈ ]0, 1[ ×[1, +∞] and (θ, p) = (1, ∞). If X = Y , then K(t, x) = min{t, 1} x . Therefore, as one can expect, (X, X)θ,p = (X, X)1,∞ = X for 0 < θ < 1, 1 ≤ p ≤ ∞, and x (X,X)θ,p 1 pθ(1 − θ) = x (X,X)θ,∞ 1/p = x x X, X , 0 < θ < 1, p < ∞, 0 < θ ≤ 1. Some inclusion properties are stated below. 3 For 0 < θ < 1, 1 ≤ p1 ≤ p2 ≤ ∞ we have Y ⊂ (X, Y )θ,p1 ⊂ (X, Y )θ,p2 ⊂ (X, Y )θ ⊂ (X, Y )θ,∞ ⊂ Y . 5) For 0 < θ1 < θ2 ≤ 1 we have (X, Y )θ2 ,∞ ⊂ (X, Y )θ1 ,1 .