{VERSION 2 3 "APPLE_PPC_MAC" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 63 "Chapter 1, Section 1: Geometry and One Type of Linear Function" }}{PARA 0 "" 0 "" {TEXT -1 51 " \+ This section begins to build intuition about " }{XPPEDIT 18 0 "L^2" " *$%\"LG\"\"#" }{TEXT -1 248 "([0,1]). Read the text for discussions of the space, the dot product, and the norm. Having a norm induces a not ion of convergence. There are the two notions of convergence other tha n norm convergence: pointwise and uniform. Here is an illustration." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "f:=(n,x)->n*x*exp(-n*x);\np lot(\{f(1,x),f(2,x),f(3,x),f(4,x),f(5,x)\},x=0..1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 167 " This sequen ce converges pointwise on [0,1] to 0 but not uniformly. The following \+ computations illustrate that this sequence of functions converges in t he norm of " }{XPPEDIT 18 0 "L^2" "*$%\"LG\"\"#" }{TEXT -1 13 "([0,1]) also." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "int(f(n,x)^2,x=0.. 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(\",n=infinity) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 " \+ Our interest in this chapter will be in solving equations of this t ype: given K and f, find y such that" }}{PARA 0 "" 0 "" {TEXT -1 32 " \+ " }{XPPEDIT 18 0 "y(x)=int(K(x,t)*y(t), t=0..1)+f(x)" "/-%\"yG6#%\"xG,&-%$intG6$*&-%\"KG6$F&%\"tG\"\"\"-F$6#F/ F0/F/;\"\"!F0F0-%\"fG6#F&F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "We require that K is in " }{XPPEDIT 18 0 "L^2" "*$%\"LG\"\"#" } {TEXT -1 31 "([0,1]x[0,1]) and that f is in " }{XPPEDIT 18 0 "L^2" "*$ %\"LG\"\"#" }{TEXT -1 57 "([0,1]). We will write this equation in the \+ symbolic form" }}{PARA 0 "" 0 "" {TEXT -1 36 " \+ y = " }{TEXT 256 1 "K" }{TEXT -1 8 "[y] + f." }}{PARA 0 "" 0 " " {TEXT -1 91 " In a matrix setting, we know how to make A* from A . Here, we illustrate the kernel of " }{TEXT 257 1 "K" }{TEXT -1 5 " a nd " }{TEXT 258 2 "K*" }{TEXT -1 23 " by drawing two graphs." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "K:=(x,t)->piecewise( x < t, \+ (x-t)^2,0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot3d(K(x,t ),x=0..1,t=0..1,axes=NORMAL);\nplot3d(K(t,x),x=0..1,t=0..1,axes=NORMAL );" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 34 "Exercises for Chapter 1, Section 1" }}{PARA 0 "" 0 "" {TEXT -1 26 "Exercise 1(a) and (b) for " }{XPPEDIT 18 0 "L^2" "*$%\"LG\"\"#" } {TEXT -1 8 "([0,1])." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "sqrt (Int((sin(Pi*x)-cos(Pi*x))^2,x=0..1))=sqrt(int((sin(Pi*x)-cos(Pi*x))^2 ,x=0..1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "arccos(Int(s in(Pi*x)*cos(Pi*x),x=0..1)/\nsqrt((Int(sin(Pi*x)^2,x=0..1)*Int(cos(Pi* x)^2,x=0..1))))=\narccos(int(sin(Pi*x)*cos(Pi*x),x=0..1)/\nsqrt((int(s in(Pi*x)^2,x=0..1)*int(cos(Pi*x)^2,x=0..1))));" }}}{PARA 0 "" 0 "" {TEXT -1 10 "Exercise 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "K :=(x,t)->1+2*x*t^2;\ny:=x->3-x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "int(K(x,t)*y(t),t=0..1);\nint(K(t,x)*y(t),t=0..1);" }}}{PARA 0 "" 0 "" {TEXT -1 10 "Exercise 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "K:=(x,t)->piecewise(x " 0 "" {MPLTEXT 1 0 84 "assume(x > 0); additionally(x<1);\nint(K (x,t)*y(t),t=0..1);\nint(K(t,x)*y(t),t=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0" 63 }{VIEWOPTS 1 1 0 1 1 1803 }