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Advanced Calculus, Fifth EditionAn inequality such as (9.61) holds for each cldsed rectangular region in D; one saysthat the F; satisfies local Lipschitz conditions in D. It is these conditions that areused in the proof rather than the hypothesis that the aFi/axj are continuous.Since the Fj are continuous, we can also choose M so that for i = 1, . . . , n,We now choose h so thatkkn 'O
Chapter 9 Ordinary Differential Equations 653by (9.6 1 ). Proceeding inductively, we find thatThe series of constantsMN converges (by the ratio test), and hence the seriesconverges uniformly for It - 71 a h to a continuous function 4i(r) (Section 6.13).The Nth partial sum of the series isHence the sequence 4,'N'(t) converges uniformly to 4,(t). Since the graph of xi =4,'N'(t) (i = 1, . . . , n) is in R for each N, we conclude that the graph of xi = #,(t)(i = 1, ..., n)isin R.Furthermore, by (9.61),16 (t, 4iN)(t), ... 9 4AN)(t))- F,(t, 4l(t), - 9 4n(t))/ 5 K 14y)(t) - 4J(t)l.,=IAccordingly, for each i the sequence F, (t, +iN'(t), . . . , +AN'(t)) converges uniformlyto F,(t, +l(t), . . . , @,(t)). Hence we can let N + oo in Eq. (9.65) to obtainnAccordingly, for It - 71 < h,and #,(i) = 2,. Thus x, = &(t) (i = 1, . . . , n) is the solution sought.11. (Linear case). Here aF,/ax, = a,,(?). By continuity we can choose K sothat Ja,,(t)l 5 K for a _( t 5 b. Thus (9.61) holds in the infinite closed regionR:a(t~b, -oo
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ContentsVectors and Matrices1.1 Int
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3.6 Combined Operations 183*3.7 Cur
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*6.24 Principal Value of Improper I
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Case of Two Particles 662Case of N
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Differential Calculusof Functionsof
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Chapter 2 Differential Calculus of
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Vector Integral CalculusPART I.TWO-
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Chapter 5 Vector Integral Calculus
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Infinite Series6.1 INTRODUCTIONAn i
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for when x = $, the numerator Ixn 1
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Fourier Series andOrthogonal Functi
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y the Schwarz inequality (7.46). By
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Functions of aComplex VariableWe as
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