A brief course in the calculus, by William Cain. by Michigan Historical Reprint Series

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Extra info for A brief course in the calculus, by William Cain.

Example text

Also make an anomalous use of g/u, g/v, g/uv, g uv etc. as designating functions with values , uv g(u,v) etc. \ respectively. Y 5. ^ ! P LO' oj The interval \uv 9 , y 1 LO+,, + J \ A xy (0 < x TT) is the Cartesian product (0,x] Co < y X(0,y]. IT) ' III. THE PRECHET VARIATION 55 -o,,, fx ,, L, P , L2 P , L' J iv y -x pT A TT -yl f 1 , "TT LX f -yl ,y _\ J pTT -x - o(x); ^y^\ P y\ f 1 * oj \u , *xfj > |_* - o(y) A \^xy I^J O (1) T(? ; -x ,TT -y] i , Lp P Lkx P B;^ , 1"xy J ley -x, If f L. -jF] P Lo>^ Jiv^y^j , i 1 /, m\ p [Tx- ; S] {** (f )] .

J, j , . , k, But now take a sequence of such gradient fields say. ) in norm, and the functions 'i converge in |f. U J F^ (x) themselves also converge towards some continuous function P(x), uniformly in every S', say. (x) with continuous derivatives in all of S, but vanishing outside some neighborhood in S each, we have the DuBois Reymond relations Inclenr-n^ent of towards + f .

5. Soc. son, "Uniformly convex spaces," Trans Compare J. A. Clark - IKD (1936), pp. Amer. Math. II. , x- then we again have h = h throughout. Finally we note that these /V -statements also hold for the modified 1 norm <$ s ^ 2 f 2)P/2 * which is structurally closer to 7 ) than (95) is. 2 it was shown by a factor the value from thet the Dirichlet Integral (20) has apart Remarks on the Problem 1. Douglas (91 2TT For k Plateau. of. 2TT - (95 \ ) ^ sin 2 ded

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