Advanced calculus; . inwhich the successive branches of fix, y) differ by a constantamount as in the case z = tan-1 (y/x) where 2 tt is the differ-ence between successive values of z for the same values of the variables. If now a circuit such as A BCB A be considered, where it is imaginedthat the origin lies within BCB, it is clear that the values of z along AB andalong BA differ by 2 ir, and whatever z gains on passing from A toB will be lost on passing from B to A, although the values throughwhich z changes will be different in the two cases by the amount2 7r. Hence the circuit ABCBA gives t
Advanced calculus; . inwhich the successive branches of fix, y) differ by a constantamount as in the case z = tan-1 (y/x) where 2 tt is the differ-ence between successive values of z for the same values of the variables. If now a circuit such as A BCB A be considered, where it is imaginedthat the origin lies within BCB, it is clear that the values of z along AB andalong BA differ by 2 ir, and whatever z gains on passing from A toB will be lost on passing from B to A, although the values throughwhich z changes will be different in the two cases by the amount2 7r. Hence the circuit ABCBA gives the same changes for z asthe simpler circuit BCB. In other words the result is obtainedthat if the different values of a multiple valued function for the samevalues of the variables differ by a constant independent of the values ofthe variables, any circuit may be reduced to circuits about the bound-aries of the portions removed; in this case the lines going from the point A to theboundaries and back may be EXERCISES 1. Draw the contour lines and sketch the surfaces corresponding tox + V „/a A^ a /m „ vy (a) z = 2(0,0) = 0, (/3) z(0, 0) = 0. x — y • ? v x + y Note that here and in the text only one of the contour lines passes through theorigin although an infinite number have it as a frontier point between two partsof the same contour line. Discuss the double limits lim Km z, lim lim z. 2. Draw the contour lines and sketch the surfaces corresponding to (a) z = x2 + y< (0) 2 (7) Z x2 + 2y*-l 2y x 2x2 + y2 — 1 Examine particularly the behavior of the function in the neighborhood of theapparent points of intersection of different contour lines. Why apparent ? 3. State and prove for functions of two independent variables the generaliza-tions of Theorems 6-11 of Chap. II. Note that the theorem on uniformity is provedfor two variables by the application of Ex. 9, p. 40, in almost the identical manneras for the case of one variable. 4. Outline definitions and t
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