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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 "" 3 256 1 {CSTYLE "" -1 -1 "" 1 14 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 "" 0 257 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 256 "" 0 "" {TEXT -1 30 "Let's see how we can do \+ on my " }{TEXT 256 22 "Modern Optics Pre-Quiz" }{TEXT -1 25 " with Map le's assistance." }}{PARA 257 "" 0 "" {TEXT -1 19 "Original Version: \+ " }{TEXT 360 15 "Earl C. Swallow" }{TEXT -1 10 " 9/18/98" }}{PARA 258 "" 0 "" {TEXT -1 33 "Revised slightly: ECS 10/11/99" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 103 "First, please recognize that this worksheet is \"swatt ing a gnat with a cannon.\" You should not really " }{TEXT 346 4 "nee d" }{TEXT -1 81 " Maple to do any of these calculations. I designed th em to be quite simple to do " }{TEXT 347 2 "if" }{TEXT -1 614 " you un derstand the relevant mathematical concepts and procedures. The purpo se of this exercise is to demonstrate the Maple procedures for doing t hese basic calculations where comparisons can easily be made to calcul ations done \"by hand\" (really, by brain). I found various Maple tra ps and confusions while preparing this worksheet. Some of them have b een left in for you to see, and others have given rise to notes includ ed in the worksheet. The best way to learn from this worksheet is to \+ use it as a template for doing the same calculations with different fu nctions which are significantly more complicated." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 2 "1." }{TEXT -1 6 " For " }{XPPEDIT 19 1 "f(x) = 5*x^2*(3 - 2*x) " "/-%\"fG6#%\"xG*(\"\"&\"\"\"*$F&\"\"#F),&\"\"$F)*&F+F)F&F)!\"\"F)" } {TEXT -1 9 ", obtain " }}{PARA 0 "" 0 "" {TEXT -1 57 " \+ (a) the first derivative of " }{TEXT 260 4 "f(x)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 58 " \+ (b) the second derivative of " }{TEXT 361 4 "f(x)" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 58 " (c) the defi nite integral of " }{TEXT 362 4 "f(x)" }{TEXT -1 13 " from 0 to 1." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "I begin w ith a " }{TEXT 257 7 "restart" }{TEXT -1 85 " for housekeeping purpose s as I re-execute the steps, and make sure Maple knows that " }{TEXT 258 1 "x" }{TEXT -1 9 " is real." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "assume(x,real);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 37 "(a) First define an EXPRESSION named " }{TEXT 363 1 "f" }{TEXT -1 108 ", then take its derivative (also defining a label for it), and then try to make it look (reasonable) pretty." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := (5*x^2)*(3 - 2*x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "D1f := diff(f,x); " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "D1f := factor(D1f);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "(b ) Now do second derivative " }{TEXT 364 6 "either" }{TEXT -1 41 " by d ifferentiating the first derivative " }{TEXT 365 2 "or" }{TEXT -1 20 " by differentiating " }{TEXT 259 1 "f" }{TEXT -1 7 " twice." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "D2fa := diff(D1f,x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "D2fb := diff(f,x$2);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "I can't get Maple to make this loo k any better, so \"by hand\" (really, by brain), I get " }{TEXT 262 18 "D2fb = 30(1 - 2x)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "(c) Now compute the definite integral of " }{TEXT 263 1 "f" }{TEXT -1 2 ", " }{TEXT 366 5 "first" }{TEXT -1 15 " the eas y way, " }{TEXT 367 4 "then" }{TEXT -1 14 " the hard way." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "I1fa := int(f,x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "I1fb := subs(x=1, int(f,x)) - subs( x=0, int(f,x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 " -- ----------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 2 "2." }{TEXT -1 5 " For " }{XPPEDIT 19 1 "g(t) = \+ A*exp(-bt)" "/-%\"gG6#%\"tG*&%\"AG\"\"\"-%$expG6#,$%#btG!\"\"F)" } {TEXT -1 6 " with " }{TEXT 368 3 "b>0" }{TEXT -1 9 ", obtain " }} {PARA 0 "" 0 "" {TEXT -1 58 " (a) the fir st derivative of " }{TEXT 264 4 "g(t)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 59 " (b) the second derivat ive of " }{TEXT 369 4 "g(t)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 59 " (c) the definite integral of " } {TEXT 370 4 "g(t)" }{TEXT -1 21 " from 0 to infinity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "assume(b > 0, A, real, t, real);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "(a) First define an EXPRESSION named g, then take its der ivative, and try to make it pretty ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "g := A*exp(-b*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "D1g := diff(g,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "That's as pretty as it gets already." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "(b) Now do second derivative " }{TEXT 371 6 "either" }{TEXT -1 41 " by differentiating the first deri vative " }{TEXT 372 2 "or" }{TEXT -1 20 " by differentiating " }{TEXT 266 1 "g" }{TEXT -1 7 " twice." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "D2ga := diff(D1g,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "D2gb := diff(g,t$2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Aga in, that's as pretty as it gets." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 64 "(c) Now integrate g once, first the easy \+ way, then the hard way." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " I1ga := int(g,t=0..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "I1gb := subs(t=infinity, int(g,t)) - subs(t=0, int(g,t));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "I1gb := eval(I1gb);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Ugh!! Maple doesn't want to acknowledge that exp(-infinity) = 0 . Try " }{TEXT 375 5 "limit" }{TEXT -1 12 " instead of " }{TEXT 373 10 "substitute" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "I1gb := limit(int(g,t), t=infinity) - subs(t=0, int(g,t));" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "I1gb := eval(I1gb);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "At last, we have it. In this case , the hard way was a lot harder, or at least more work." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 " \+ -----------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 322 1 "3" }{TEXT -1 15 ". What are the " } {TEXT 376 4 "real" }{TEXT -1 5 " and " }{TEXT 377 9 "imaginary" } {TEXT -1 10 " parts of " }{XPPEDIT 19 1 "exp(i phi)" "-%$expG6#*&%\"iG \"\"\"%$phiGF'" }{TEXT -1 4 "? [" }{TEXT 270 8 "Remember" }{TEXT -1 42 " that Maple uses I = sqrt(-1) rather than " }{TEXT 269 1 "i" } {TEXT -1 2 ".]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "assume(phi, real);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "z := exp(I*phi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "realpart := Re(z); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "imaginarypart := Im(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Note that the " }{TEXT 271 14 "imaginary \+ part" }{TEXT -1 6 " is a " }{TEXT 272 4 "real" }{TEXT -1 74 " quantity -- a strange, but standard, convention of mathematical language." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 " \+ -------------------------------------------- ----------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Now it's " }{TEXT 293 6 "VECTOR" }{TEXT -1 6 " time!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 321 1 "4" } {TEXT -1 16 ". Given vectors " }{TEXT 273 1 "A" }{TEXT -1 5 " = 2 " } {TEXT 274 1 "i" }{TEXT -1 3 " - " }{TEXT 275 1 "j" }{TEXT -1 3 " + " } {TEXT 276 1 "k" }{TEXT -1 5 " and " }{TEXT 277 1 "B" }{TEXT -1 3 " = \+ " }{TEXT 278 1 "i" }{TEXT -1 4 " + 3" }{TEXT 279 1 "j" }{TEXT -1 4 " - 2" }{TEXT 280 1 "k" }{TEXT -1 11 ", evaluate " }}{PARA 0 "" 0 "" {TEXT -1 30 " (a) " }{TEXT 281 1 "A" }{TEXT -1 3 " + " }{TEXT 282 1 "B" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 30 " (b) " }{TEXT 283 1 "A" }{TEXT -1 3 " \+ - " }{TEXT 284 1 "B" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 30 " \+ (c) " }{TEXT 285 1 "A" }{TEXT -1 1 " " } {TEXT 291 3 "dot" }{TEXT -1 1 " " }{TEXT 286 1 "B" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 30 " (d) " }{TEXT 287 1 "A" }{TEXT -1 1 " " }{TEXT 292 5 "cross" }{TEXT -1 1 " " }{TEXT 288 1 "B" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 31 " \+ (e) |" }{TEXT 289 1 "A" }{TEXT -1 7 "| and |" }{TEXT 290 1 "B" }{TEXT -1 7 "|, and " }}{PARA 0 "" 0 "" {TEXT -1 34 " \+ (f) the " }{TEXT 378 6 "cosine" }{TEXT -1 22 " of t he angle between " }{TEXT 300 1 "A" }{TEXT -1 5 " and " }{TEXT 301 1 " B" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "Start by loading linear algebra \"stuff\" (library packag e) for working with vectors, " }{TEXT 379 4 "etc." }{TEXT -1 14 " The n define " }{TEXT 294 1 "A" }{TEXT -1 5 " and " }{TEXT 295 1 "B" } {TEXT -1 19 " and start to work." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A := vect or([2, -1, 1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "B := vec tor([1, 3, -2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 60 "First, two alternatives to (a) add and (b) subtrac t vectors " }{TEXT 296 1 "A" }{TEXT -1 5 " and " }{TEXT 297 2 "B." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ABsum := evalm(A + B);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "or" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ABsum2 := matadd(A,B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ABdifference := evalm(A - B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "or" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ABsu m2 := matadd(A,-B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Now (c) dot and (d) cross vectors " }{TEXT 298 1 "A" }{TEXT -1 5 " and " }{TEXT 299 1 "B" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ABdot := dotprod(A,B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "For fun, check that the dot product is th e same in either order," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " BAdot := dotprod(B,A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "A Bcross := crossprod(A,B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "and \+ verify that the cross product is not (it \"anti-commutes\")." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "BAcross := crossprod(B,A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Note that " }{TEXT 302 1 "B" } {TEXT -1 1 " " }{TEXT 307 5 "cross" }{TEXT -1 1 " " }{TEXT 303 1 "A" } {TEXT -1 5 " = - " }{TEXT 304 1 "A" }{TEXT -1 1 " " }{TEXT 306 5 "cros s" }{TEXT -1 1 " " }{TEXT 305 1 "B" }{TEXT -1 50 ", as required by th e definition of cross product." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 35 "Finally, get (e) the magnitudes of " } {TEXT 308 1 "A" }{TEXT -1 5 " and " }{TEXT 309 1 "B" }{TEXT -1 12 " as well as " }}{PARA 0 "" 0 "" {TEXT -1 12 "(f) the the " }{TEXT 380 6 " cosine" }{TEXT -1 45 " of the angle between them from cos(theta) = " } {TEXT 312 1 "A" }{TEXT -1 1 " " }{TEXT 316 3 "dot" }{TEXT -1 1 " " } {TEXT 313 1 "B" }{TEXT -1 3 "/(|" }{TEXT 314 1 "A" }{TEXT -1 3 "| |" } {TEXT 315 1 "B" }{TEXT -1 3 "|)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "modA := norm(A, frobenius);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 2 "or" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "modA2 : = sqrt(dotprod(A,A));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "To get a decimal value for |" }{TEXT 310 1 "A" }{TEXT -1 29 "|, if you happen \+ to want one:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "modAf := ev alf(modA);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "modB := norm( B, frobenius);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "or" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "modB2 := sqrt(dotprod(B,B));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "To get a decimal value for |" } {TEXT 311 1 "B" }{TEXT -1 29 "|, if you happen to want one:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "modBf := evalf(modB);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "No w work on the angle between " }{TEXT 317 1 "A" }{TEXT -1 5 " and " } {TEXT 318 1 "B" }{TEXT -1 41 ". First compute the cosine of the angle ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "cosAB := ABdot/(modA*m odB);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Now clean it up to a bit simpler form or to decimal form if desired." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "cosAB := simplify(cosAB);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "cosABf := evalf(cosAB);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Might as well go ahead and get the angle, just for f un, though the question did " }{TEXT 319 3 "not" }{TEXT -1 12 " ask fo r it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "thetaAB := arccos( cosAB);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "thetaABf := eval f(thetaAB);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "A Maple shortcut f or the angle would be:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "t hetaAB2 := angle(A, B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " thetaABb2 := evalf(thetaAB2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 " The angle is in radians. Let's convert to degrees \"by hand\" and wit h Maple function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "thetaA Bdeg := evalf(thetaAB*180/Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "thetaABdegb := evalf(convert(thetaAB, degrees));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 " -------- ----------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 320 2 "5." }{TEXT -1 5 " For " }{XPPEDIT 19 1 "F(x,y) \+ = 6 x^2 y^3 + 3y + 92" "/-%\"FG6$%\"xG%\"yG,(*(\"\"'\"\"\"*$F&\"\"#F+F '\"\"$F+*&F.F+F'F+F+\"##*F+" }{TEXT -1 9 ", obtain " }}{PARA 0 "" 0 " " {TEXT -1 61 " (a) the partial derivati ve of " }{TEXT 324 6 "F(x,y)" }{TEXT -1 17 " with respect to " }{TEXT 323 1 "x" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 61 " \+ (b) the partial derivative of " }{TEXT 382 6 "F(x,y )" }{TEXT -1 17 " with respect to " }{TEXT 325 1 "y" }{TEXT -1 6 ", an d " }}{PARA 0 "" 0 "" {TEXT -1 68 " (c) \+ the second partial derivative of " }{TEXT 328 1 "F" }{TEXT -1 17 " wit h respect to " }{TEXT 327 1 "y" }{TEXT -1 11 ", and then " }{TEXT 329 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "F(x,y) := 6*(x^2)*(y^3) + 3*y + 92;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "dFdx := diff(F(x,y),x); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dFdy := diff(F(x,y),y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "d2Fdxdy := diff(dFdx,y);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "or" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "d2Fdxdyb := diff(F(x,y),x,y);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "and, to show that order of differentiation does " } {TEXT 330 3 "not" }{TEXT -1 19 " change the result:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "d2Fdydx := diff(dFdy,x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 " \+ ---------------------------" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 331 2 "5." }{TEXT -1 32 " For the vector valued function " }{TEXT 332 1 "E" }{TEXT -1 13 "(x,y, z) = xy " }{TEXT 333 1 "i" }{TEXT -1 6 " + xz " }{TEXT 334 1 "j" } {TEXT -1 6 " + yz " }{TEXT 335 1 "k" }{TEXT -1 11 ", evaluate " }} {PARA 0 "" 0 "" {TEXT -1 53 " (a) the di vergence of " }{TEXT 336 2 "E," }{TEXT -1 5 " and " }}{PARA 0 "" 0 "" {TEXT -1 47 " (b) the curl of " }{TEXT 337 1 "E" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 338 4 "Note" }{TEXT -1 22 ": it appears that my " } {TEXT 339 7 "restart" }{TEXT -1 35 " at this point wipes out access to " }{TEXT 340 6 "linalg" }{TEXT -1 24 ". Thus, I have to do a " } {TEXT 348 12 "with(linalg)" }{TEXT -1 12 " again here." }}{PARA 0 "" 0 "" {TEXT -1 174 "I choose to work this way so the computations for e ach problem will be independent and free-standing. This avoids potent ial confusions and surprises. Also, I use a : after " }{TEXT 383 12 " with(linalg)" }{TEXT -1 55 " so that it won't list all of its contents this time. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "E := vector([x*y, x*z, y*z]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 341 4 "Note" }{TEXT -1 12 " that it is " }{TEXT 342 9 "essential" }{TEXT -1 28 " to include the \"*\" in x*y, \+ " }{TEXT 384 3 "etc" }{TEXT -1 59 ". Otherwise, Maple thinks it is a \+ new quantity named \"xy\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "r := vector([x, y, z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "divE := diverge(E, r);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "One can also do the divergence \"by hand\" as follows." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "divEb := diff(E[1], x) + diff(E[2], y) + \+ diff(E[3], z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "curlE := \+ curl(E, r);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "No way am I going to struggle through doing the curl by hand. Of course, it's quite po ssible to do it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 " -------------- ----------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 343 2 "7." }{TEXT -1 5 " For " }{XPPEDIT 19 1 "V(x,y,z) = \+ x^2 + x*y + x*z" "/-%\"VG6%%\"xG%\"yG%\"zG,(*$F&\"\"#\"\"\"*&F&F,F'F, F,*&F&F,F(F,F," }{TEXT -1 9 ", obtain " }}{PARA 0 "" 0 "" {TEXT -1 54 " (a) the gradient of " }{TEXT 345 2 "V," }{TEXT -1 5 " and " }}{PARA 0 "" 0 "" {TEXT -1 52 " \+ (b) the Laplacian " }{TEXT 344 1 "V" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(li nalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "V := x^2 + x*y + \+ x*z;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "r := vector([x, y, \+ z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "gradV := grad(V,r); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "or, \"by hand\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "gradVb := vector([diff(V,x),diff(V, y),diff(V,z)]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 " ------- --------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 349 2 "8." }{TEXT -1 43 " Show that complex harmonic wa ve function " }{XPPEDIT 19 1 "U(x,t) = A*exp(i*(k*x - omega*t))" "/-% \"UG6$%\"xG%\"tG*&%\"AG\"\"\"-%$expG6#*&%\"iGF*,&*&%\"kGF*F&F*F**&%&om egaGF*F'F*!\"\"F*F*" }{TEXT -1 54 " is a solution of the simple class ical wave equation " }{XPPEDIT 19 1 "diff(U(x,t),x,x) = diff(U(x,t),t, t)/v^2" "/-%%diffG6%-%\"UG6$%\"xG%\"tGF)F)*&-F$6%-F'6$F)F*F*F*\"\"\"*$ %\"vG\"\"#!\"\"" }{TEXT -1 18 " if and only if " }{XPPEDIT 19 1 "v = omega/k" "/%\"vG*&%&omegaG\"\"\"%\"kG!\"\"" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "assume(k > 0, omega > 0, v > 0, x, real, t, real);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Uh := A*exp(I*(k *x - omega*t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "waveeq : = diff(U(x,t),x$2) = (1/v^2)*diff(U(x,t),t$2);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "subs(U(x,t) = Uh, waveeq);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vsols := solve(\",v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "v := vsols[1];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "QED" }}{PARA 0 "" 0 "" {TEXT -1 104 "Note that the solutions have two signs, corresponding to waves goint to the right (+) or to the left(- )." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 " \+ ---------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 350 2 "9." }{TEXT -1 28 " For tw o given 2x2 matrices " }{TEXT 351 1 "G" }{TEXT -1 5 " and " }{TEXT 352 1 "H" }{TEXT -1 34 ", and a given (2x1) column matrix " }{TEXT 353 1 "V" }{TEXT -1 11 ", evaluate " }}{PARA 0 "" 0 "" {TEXT -1 36 " \+ (a) " }{TEXT 354 2 "GH" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 36 " (b) \+ " }{TEXT 355 2 "HG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 36 " \+ (c) " }{TEXT 356 2 "GV" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 53 " (d) t he transpose of " }{TEXT 357 1 "G" }{TEXT -1 6 ", and " }}{PARA 0 "" 0 "" {TEXT -1 55 " (e) the determinant \+ of " }{TEXT 358 1 "G" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Below I employ \+ " }{TEXT 359 3 "two" }{TEXT -1 87 " of the several alternatives for cr eating a matrix (just to show how they can be used)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "G := matrix(2,2,[1,2,3,4]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "H := matrix([[5,3],[2,1]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "V := matrix([[2],[6]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "GH := multiply(G,H);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "HG := multiply(H,G);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "As expected, " }{TEXT 385 2 "GH" }{TEXT -1 4 " is " } {TEXT 386 3 "not" }{TEXT -1 10 " equal to " }{TEXT 387 2 "HG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "GV := multiply(G ,V);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "GT := transpose(G); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "detG := det(G);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "At last, we are at the end. Now w asn't that fun?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 3 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 }