By Serge Lang

ISBN-10: 0201042991

ISBN-13: 9780201042993

This can be a new, revised version of this well known textual content. the entire simple subject matters in calculus of a number of variables are lined, together with vectors, curves, capabilities of numerous variables, gradient, tangent aircraft, maxima and minima, power 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 incorporated.

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**Extra resources for Calculus of Several Variables**

**Example text**

We consider examples of boundary-value problems in one and two space variables. EXAMPLES WITH WAVE EQUATION. (1) A string on the x axis under tension is such that each point can be displaced only in the y direction. Let y = u(x, t) be the displacement. The equation for the unknown function u(x, t) in suitable physical units is u tt = u x x , and the boundary data are u(x, 0) = f (x) u t (x, 0) = g(x) u(0, t) = u(l, t) = 0 (initial displacement), (initial velocity), (ends of string ﬁxed for all t ≥ 0).

59 of Basic. I. Introduction to Boundary-Value Problems 24 a countable subcover. Since the sets B(k; ϕα ) are disjoint, we conclude that the set of all ϕα is countable. Hence E is at most countably inﬁnite. The next step is to bound E below under additional hypotheses as in the statement of the theorem. Let λ be in E, and let ϕ be a nonzero solution of (SL) corresponding to λ and normalized so that ϕ r = 1. Multiplying (SL1) by ϕ¯ and integrating, we have b λ= b λ|ϕ|2r dt = − a = − pϕ ϕ¯ a b a b + a b ( pϕ ) ϕ¯ dt + q|ϕ|2 dt a b p|ϕ |2 dt + q|ϕ|2 dt a b ≥ − p(b)ϕ (b)ϕ(b) + p(a)ϕ (a)ϕ(a) + (|ϕ|2r )(r −1 q) dt a ≥ − p(b)ϕ (b)ϕ(b) + p(a)ϕ (a)ϕ(a) + inf {r (t)−1 q(t)}.

Hence there is a nonzero vector u with L(u) = cu for some real c. Normalizing, we may assume that u has norm 1. If M consists of the scalar multiples of u, then L carries M ⊥ to itself, and the restriction of L to M ⊥ is self adjoint. Proceeding inductively, we obtain a system of orthogonal eigenvectors for L, each of norm 1. A certain amount of this argument works in the inﬁnite-dimensional case. In fact, suppose that L is self adjoint. Then any u in H has (L(u), u) = (u, L ∗ (u)) = (u, L(u)) = (L(u), u), and hence the function u → (L(u), u) is real-valued.

### Calculus of Several Variables by Serge Lang

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