InterchangeDiffandIntegral

InterchangeDiffandIntegral - 2005-06 Second Term Notes...

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Unformatted text preview: 2005-06 Second Term MAT2060B1Supplementary Notes 3Interchange of Differentiation and IntegrationThe theme of this course is about various limiting processes. We have learntthe limits of sequences of numbers and functions, continuity of functions, limits ofdifference quotients (derivatives), and even integrals are limits of Riemann sums.As often encountered in applications, exchangeability of limiting processes is animportant topic. For example, we learntddxZxaf=Zxadfdx,f(a) = 0,wheneverdfdxis integrable; alsolimn→∞ddxfn(x) =ddxlimn→∞fn(x),if{fn}and{fn}converge uniformly.Here we consider the following situation. Letf(x, y) be a function defined in[a, b]×[c, d] andφ(y) =Zbaf(x, y)dx.It is natural to ask if continuity and differentiability are preserved under integration.Theorem 1.Letf(x, y)be continuous in[a, b]×[c, d]. Thenφdefined above is acontinuous function on[c, d].Proof.Sincefis continuous in [a, b]×[c, d], it is bounded and uniformly continuous.In other words, for anyε >0,∃δsuch that|φ(y)−φ(y)| ≤Zba|f(x, y)−f(x, y)|dx< ε(b−a)∀y,|y−y|< δ,which shows thatφis uniformly continuous on [c, d].Source: http://www.math.cuhk.edu.hk/~mat2060/mat2060b/Notes/notes3.pdf2005-06 Second Term MAT2060B2Theorem 2.Letfand∂f∂ybe continuous in[a, b]×[c, d]. Thenφis differentiableandddyφ(y) =Zba∂f∂y(x, y)dxholds.Proof.Fixy∈(c, d),y+h∈(c, d) for smallh∈R,φ(y+h)−φ(y)h=1hZba(f(x, y+h)−f(x, y))dx=Zba∂f∂y(x, z)dxwherezis a point betweenyandy+hwhich depends onx. In any case,ÿÿÿφ(y+h)−φ(y)h−Zba∂f∂y(x, y)dxÿÿÿ≤Zbaÿÿÿ∂f∂y(x, z)−∂f∂y(x, y)ÿÿÿdx.Since∂f∂yis uniformly continuous on [a, b]×[c, d], forε >0,∃δsuch thatÿÿÿ∂f∂y(x, y)−∂f∂y(x, y)ÿÿÿ< ε,∀|y−y|< δand∀x.Takingh≤δ, we getÿÿÿφ(y+h)−φ(y)h−Zba∂f∂y(x, y)dxÿÿÿ< ε,whence the condition follows.Wheny=cord, the same proof works with some trivial changes.In many applications, the rectangle is replaced by an unbounded region. Whenthis happens, we need to consider improper integrals. As a typical case, let’s assumefis defined in [a,∞)×[c, d] and setφ(y) =Z∞af(x, y)dx.2005-06 Second Term MAT2060B3The functionφ(y) makes sense if the improper integralZ∞af(x)dxis well-definedfor eachy. Recall that this meanslimb→∞Zbaf(x, y)dxexists. We introduce the following definition: The improper integralZ∞af(x, y)dxis uniformly convergent if∀ε,∃b>0 such thatÿÿÿZbbf(x, y)dxÿÿÿ< ε,∀b, b≥b....
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InterchangeDiffandIntegral - 2005-06 Second Term Notes...

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