{VERSION 2 3 "SUN SPARC SOLARIS" "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 "" -1 264 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 333 "" 0 1 0 0 0 0 0 0 1 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 "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet It em" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 } {PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 49 "Change of Variables and A pplications of Integrals" }}{PARA 258 "" 0 "" {TEXT 264 12 "Chapter 6, " }{TEXT 265 15 "Vector Calculus" }{TEXT 266 18 ", Marsden & Tromba " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 28 "Pre pared by Douglas B. Meade" }}{PARA 257 "" 0 "" {TEXT -1 13 " 6 March 1 997" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "with( linalg ); # linear algebra routines (e.g. Jacob ian and determinant)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "wit h( plots ); # special plotting routines" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "setoptions( labels=[`x`,`y`] ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "setoptions3d( labels=[`x`,`y`,`z`] ):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "with( student ); \+ # extra tools for one-variable calculus" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "libname := `/m1/faculty/meade/computer/symbolic/ma ple/lib-mVr4/mvcal2/`, libname:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 " # valid only on USC Math network" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "with( mvcal2 ); # e xtra tools for 2- and 3-d calculus" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 39 "Section 6.1: The Geometry of Maps from " }{XPPEDIT 18 0 "R^2" "*$%\"RG\"\"#" } {TEXT -1 4 " to " }{XPPEDIT 18 0 "R^2" "*$%\"RG\"\"#" }{TEXT -1 1 " " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Notation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "D" "I\"DG6\"" }{TEXT -1 19 "* b e a subset of " }{XPPEDIT 18 0 "R^2" "*$%\"RG\"\"#" }{TEXT -1 11 " a nd T : " }{XPPEDIT 18 0 "D" "I\"DG6\"" }{TEXT -1 5 "* -> " }{XPPEDIT 18 0 "R^2" "*$%\"RG\"\"#" }{TEXT -1 20 ". The image set of " } {XPPEDIT 18 0 "D" "I\"DG6\"" }{TEXT -1 17 "* under T is " } {XPPEDIT 18 0 "D" "I\"DG6\"" }{TEXT -1 3 "=T(" }{XPPEDIT 18 0 "D" "I\" DG6\"" }{TEXT -1 3 "*)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Example 1 - Polar Coordinat es" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Dstar := theta = 0 .. \+ 2*Pi, r = 0 .. 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "var := [ r, theta ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "T := [ r* cos(theta), r*sin(theta) ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plotDstar := regionplot2d( Dstar, var, title=`D*` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plotD := regionplot2d( Dstar, T, title= `D=T(D*)` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "display( array([plo tDstar,plotD]) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Example 2 - Linear Transformation " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Dstar := x = -1 .. 1, y \+ = -1 .. 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "var := [ x, y ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "T := [ (x+y)/2, (x-y )/2 ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plotDstar := regi onplot2d( Dstar, var, title=`D*` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plotD := regionplot2d( Dstar, T, title=`D=T(D*)` ):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "display( array([plotDstar,plotD]) ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 44 "Sec tion 6.2: The Change of Variables Theorem" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Figure 6.2.1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Dstar := u = 0 .. 1, v = 0 .. 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "var := [ u, v ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "T := [ -u^2+4*u, v ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plotDstar := regionplot2d( Dstar, var, title=`D*` ): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plotD := regionplot2d( Dsta r, T, title=`D=T(D*)` ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "plotD star;plotD;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Example 1 - Jacobian determinant" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "T := [ r * cos(theta), r * s in(theta) ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "var := [ r, theta ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "j := jacobian( T, var );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "jd := det( j \+ );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "jd := simplify( jd ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 33 "Example 6 - spherical coordinates" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "T := [ rho*sin(phi)*cos(theta), rho*sin(p hi)*sin(theta), rho*cos(phi) ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "var := [ rho, theta, phi ];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "j := jacobian( T, var );" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "jd := det( j );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "jd := simplify( jd );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 56 "Section 6.3: Applications of Doub le and Triple Integrals" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "Proble m 8, p. 384 - mass and center of mass of a 3D solid" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Find the mass of the solid bounded by the cylin der " }{XPPEDIT 18 0 "x^2+y^2=2*x" "/,&*$%\"xG\"\"#\"\"\"*$%\"yG\"\"# F'*&\"\"#F'F%F'" }{TEXT -1 16 " and the cone " }{XPPEDIT 18 0 "z^2 = x^2+y^2" "/*$%\"zG\"\"#,&*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F*" }{TEXT -1 16 " with density " }{XPPEDIT 18 0 "rho=sqrt(x^2+y^2)" "/%$rhoG-%%sq rtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F+" }{TEXT -1 144 ". The mass is given by a triple integral. The key is describing the region in an ap propriate coordinate system. Note the frequent presence of " } {XPPEDIT 18 0 "x^2+y^2" ",&*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F&" }{TEXT -1 122 "; this suggests the use of cylindrical coordinates. The top o f the solid is formed by the portion of the cone above the " } {XPPEDIT 18 0 "x-y" ",&%\"xG\"\"\"%\"yG!\"\"" }{TEXT -1 10 " plane: \+ " }{XPPEDIT 18 0 "z = r" "/%\"zG%\"rG" }{TEXT -1 75 "; the bottom of \+ the solid is formed by the portion of the cone below the " }{XPPEDIT 18 0 "x-y" ",&%\"xG\"\"\"%\"yG!\"\"" }{TEXT -1 10 " plane: " } {XPPEDIT 18 0 "z = -r" "/%\"zG,$%\"rG!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The projection \+ of the solid back into the " }{XPPEDIT 18 0 "x-y" ",&%\"xG\"\"\"%\"yG! \"\"" }{TEXT -1 138 " plane is the cross-section of the cylinder. Not e that the equation of the cylinder, when written in cylindrical (pola r) coordinates is " }{XPPEDIT 18 0 "r = 2*cos(theta)" "/%\"rG*&\"\"# \"\"\"-%$cosG6#%&thetaGF&" }{TEXT -1 235 ". This is a circle (centere d at (1,0) with radius 1). To describe this in polar coordinates takes a little care. In particular, we must ensure that the radius is alway s positive. Thus, the angle theta needs to come from the interval " } {XPPEDIT 18 0 "-Pi/2" ",$*&%#PiG\"\"\"\"\"#!\"\"F'" }{TEXT -1 4 " <= \+ " }{XPPEDIT 18 0 "theta" "I&thetaG6\"" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "Pi/2" "*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Thus, the cross-sect ions can be described (in polar coordinates) as" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 53 "drdtplot( r=0..2*cos(theta), theta = -Pi/2 .. \+ Pi/2 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "When the third dimensi on is added to the problem, the solid is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "dzdrdtplot( z=-r..r, r=0..2*cos(theta), theta=-Pi/2.. Pi/2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The density is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "rho := r;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 " so the mass will be given by the triple integral (note the inclusion o f the Jacobian determinant for cylindrical coordinates):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "M := Int( Int( Int( rho * r, z = -r .. r ), r = 0 .. 2*cos(theta) ), theta = -Pi/2 .. Pi/2 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "The first two integrals are easy to evalu ate. The last (theta) integral will involve integrating " }{XPPEDIT 18 0 "cos(theta)^4" "*$-%$cosG6#%&thetaG\"\"%" }{TEXT -1 51 " -- use t he table of integrals. The final value is:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "value( M );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Note that symmetry can be used to simplify the calculations." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "The problem does not ask for the center of mass, but thi s information can be obtained with almost no additional effort. Symmet ry (see the plot) suggests the center of mass should occur in the plan es " }{XPPEDIT 18 0 "z=0" "/%\"zG\"\"!" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "y=0" "/%\"yG\"\"!" }{TEXT -1 7 ". The " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 65 "-coordinate of the center of mass shoul d occur somewhere beyond " }{XPPEDIT 18 0 "x=1" "/%\"xG\"\"\"" } {TEXT -1 80 ", but the exact location is requires the evaluation of on e more triple integral:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "xbar := Int( Int( Int( (r*cos(theta)) * rho * r, z = -r .. r ), r = 0 .. 2*cos(theta) ), theta = -Pi/2 .. Pi/2 )/M;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "value( xbar );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "The correspon ding integrals for the other components of the center of mass are:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "ybar := Int( Int( Int( (r* sin(theta)) * rho * r, z = -r .. r ), r = 0 .. 2*cos(theta) ), theta = -Pi/2 .. Pi/2 ) / M;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "va lue( ybar );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "zbar := In t( Int( Int( z * rho * r, z = -r .. r ), r = 0 .. 2*cos(theta) ), thet a = -Pi/2 .. Pi/2 ) / M;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "value( zbar );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Thus, the cent er of mass is ( 4/3, 0, 0 )." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Extensions to Problem \+ 8, p. 384" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "What are the mass and center of mass of" }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 39 "the top-ha lf of the solid in Problem 8?" }}{PARA 15 "" 0 "" {TEXT -1 66 "the por tion of the solid in Problem 8 that is in the first octant?" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "I will accept solutions to these p roblems as " }{TEXT 333 12 "extra credit" }{TEXT -1 275 ". The deadlin e for solutions is the first class period after USC losses in the 1997 NCAA basketball tournament or the end of March (1997), whichever come s last. You do not have to show all of the algebra, but you should ind icate what is done to get from one step to the next." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "1 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 }