By Leonid P. Lebedev

This publication deals a short, essentially entire, and comparatively uncomplicated advent to practical research. It additionally illustrates the appliance of practical analytic how to the technology of continuum mechanics. summary yet robust mathematical notions are tightly interwoven with actual principles within the therapy of nontrivial boundary price difficulties for mechanical objects.

This moment variation comprises extra prolonged insurance of the classical and summary parts of practical research. Taken jointly, the 1st 3 chapters now represent a typical textual content on utilized sensible research. This strength use of the e-book is supported by way of a considerably prolonged set of routines with tricks and strategies. a brand new appendix, supplying a handy directory of crucial inequalities and imbedding effects, has been added.

The e-book should still entice graduate scholars and researchers in physics, engineering, and utilized mathematics.

Reviews of first edition:

"This ebook covers useful research and its functions to continuum mechanics. The presentation is concise yet entire, and is meant for readers in continuum mechanics who desire to comprehend the mathematical underpinnings of the self-discipline. … designated strategies of the workouts are supplied in an appendix." (L’Enseignment Mathematique, Vol. forty nine (1-2), 2003)

"The reader comes away with a profound appreciation either one of the physics and its value, and of the wonderful thing about the practical analytic technique, which, in skillful arms, has the facility to dissolve and make clear those tricky difficulties as peroxide does clotted blood. various workouts … try out the reader’s comprehension at each level. Summing Up: Recommended." (F. E. J. Linton, selection, September, 2003)

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That is, S is convex if tx + (1 − t)y ∈ S for every pair of elements x, y ∈ S . 4. A set in a metric space X is bounded if there is a ball of ﬁnite radius that contains all the elements of the set. 2. (a) Show that the union of two bounded sets is a bounded set. (b) Show that the open and closed balls B(0, r) and B(0, r) are convex sets. (c) Show that if A and B are convex sets, then so is the intersection A ∩ B. 7 Convergence in a Metric Space While geometric ﬁgures in metric spaces can be interesting, we are more interested in properties that extend the usual ideas of calculus to function spaces.

We shall show that X i → X as i → ∞. We have d(X i , X) ≤ d(X i , Xi ) + d(Xi , X) ≤ = 1 + d(Xi , X) i 1 + lim d(xi , x j ) → 0 as i → ∞ i j→∞ since {xi } is a Cauchy sequence. This completes the proof of (b). 34 1 Metric, Banach, and Hilbert Spaces ˜ in M ∗ . Let X be an equivalence class containing a representa(c) Density of M tive sequence {xn }. Denoting by Xn the stationary class for the stationary sequence (xn , xn , . ), we have d(Xn , X) = lim d(xn , xm ) → 0 as n → ∞ m→∞ since {xn } is a Cauchy sequence.

3. 3). In mechanics, the work of external forces often takes the form Ω F(x)G(x) dΩ . Let us determine this integral when F(x) ∈ L p (Ω) , G(x) ∈ Lq (Ω) , 1 1 + =1. p q Consider In = Ω fn (x)gn (x) dΩ where { fn (x)} and {gn (x)} are representative sequences of F(x) and G(x), respectively. We will show that {In } is a numerical Cauchy sequence and therefore possesses a limit. We have |In − Im | = = and therefore Ω Ω { fn (x)gn (x) − fm (x)gm (x)} dΩ {[ fn (x) − fm (x)]gn (x) + fm (x)[gn (x) − gm (x)]} dΩ 38 1 Metric, Banach, and Hilbert Spaces |In − Im | ≤ Ω ≤ | fn (x) − fm (x)| |gn (x)| dΩ + Ω + | fn (x) − fm (x)| p dΩ Ω | fm (x)| p dΩ Ω 1/p Ω 1/p Ω | fm (x)| |gn (x) − gm (x)| dΩ |gn (x)|q dΩ 1/q |gn (x) − gm (x)|q dΩ 1/q → 0 as n, m → ∞ since { fn (x)} and {gn (x)} are Cauchy sequences in their respective metrics and Ω | fn (x)| p dΩ → Ω |F(x)| p dΩ , Ω |gn (x)|q dΩ → Ω |G(x)|q dΩ .