By Nikolsky S.M.

A textbook for college scholars (physicists and mathematicians) with certain supplementary fabric on mathematical physics. in accordance with the direction learn by means of the writer on the Moscow Engineering Physics Institute. quantity 2 comprises a number of integrals, box thought, Fourier sequence and Fourier essential, differential manifolds and differential kinds, and the Lebesgue indispensable.

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**Example text**

Tr examples of terms. For instance, Note that in (3) and (4) we have written and not i-ti ahhresiates the (bllowing: It, consists or,,(** followed by the expression u followed by "r" followed by the expression it, follo«ed by ")". A term containing no variables is called a closed term. An expression x is called a jurnttrla (of 2') if there is a finite sequence of c\pressions xt,x...... x - 2 such that for each i. 1 < i 5 it, either 7, - (it = r) where At and r are terms of 1'. or i. 2. or r, _ -1 xj where j < i.

Ou se. sIIICC h c S. i But for all J. h = :bI - pfg;..... rI;. h a0ain sixes the result. Ila%ine settled the cage k -I 0. t Ti k It. :he induction hypothesis. e. ,;. T . :.. 5 and the definition of''p. C. gi) . ; o , . e. g, .. e. 1 I I. 2. Let u =u(xt,, ... ,x,,,) be a term of 2', and let g', ... g" e Z. For each 3 e I, let ha =1p(gb, ... '01Then h e Z and PROOF. Assuming for the moment that h e Z, let g = I*p(g', ... ,g")l*. e. 3, Ti, the desired result. 1. If u is a constant, u = b, then h,, = b for all S and he Z; if y = x;", then h = g1 a Z.

And then write ,et(b...... b,) for the closed terns obtained by replacing each occurrence of x,, by b, x by b,, and so forth. Y' we write when all variables which ha%a free occurrences in x are included in the list x....... x,,,. In this case. bk) for the sentence obtained by replacing each free occurrence of xt, by b,. x;, by b2, and so forth. 6. SEMANTICS It is intended that each closed term of Y'L "represents" a definite element of U. and that each sentence of 2 makes an assertion. true or false, about L .