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in Maple using the command " }{TEXT 277 6 "vector" }{TEXT 276 39 ". For example, to define the vector v=2" }{TEXT 278 1 "i" } {TEXT 276 2 "+3" }{TEXT 278 1 "j" }{TEXT 276 2 "-4" }{TEXT 278 1 "k" } {TEXT 276 43 " in three-dimensional space, use the syntax" }}{PARA 208 "" 0 "" {TEXT 276 1 " " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 21 "v1:=vector([2,3,-4]);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I#v 1G6\"-I'vectorG6$I*protectedGF)I(_syslibGF%6#7%\"\"#\"\"$!\"%" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 18 "The output of the " }{TEXT 277 6 "vector" }{TEXT 276 750 " command looks like a list [2,3,-4]. Mathematically speaking, as \+ long as you remain within a fixed coordinate system, a vector can be i dentified with a list of its components. Nevertheless, to Maple, a lis t and a vector are two different objects. In most commands that we ar e going to use below, you can enter vectors as lists, and Maple will u nderstand what you mean. Later on, however, when we need more advanced commands involving vectors, the distinction between lists and vectors will become important. It is of crucial importance in Maple programmi ng. Should you try to program something yourself, you have to carefull y distinguish between vectors and lists. Incidentally, you can ask Map le what it considers a given object to be by using the " }{TEXT 277 4 "type" }{TEXT 276 64 " command. Let's see what v1 and [2,3,-4] are acc ording to Maple." }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 16 "type(v1,vector);" }}{PARA 211 "" 1 "" {XPPMATH 20 " 6#I%trueGI*protectedGF$" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 " " {MPLTEXT 1 273 14 "type(v1,list);" }}{PARA 211 "" 1 "" {XPPMATH 20 " 6#I&falseGI*protectedGF$" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 43 "type([2,3,-4],list); type([2,3,-4],vector);" }} {PARA 211 "" 1 "" {XPPMATH 20 "6#I%trueGI*protectedGF$" }{TEXT 279 1 " " }}{PARA 211 "" 1 "" {XPPMATH 20 "6#I&falseGI*protectedGF$" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 105 "As you see, Maple considers v1 to be a vector but not a list, and [2,3,-4] to be a list but not a vector." }}{PARA 208 "" 0 " " }{PARA 208 "" 0 "" {TEXT 276 57 "Most commands to manipulate vectors are contained in the " }{TEXT 277 6 "linalg" }{TEXT 276 57 " package \+ (\"linalg\" for \"linear algebra\"). Let's load the " }{TEXT 277 6 "li nalg" }{TEXT 276 123 " package. If you are curious about its content, \+ replace the colon by a semicolon on the line below before re-executing it. " }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 13 "with(linalg):" }}{PARA 212 "" 1 "" {TEXT 280 79 "Warning, the \+ protected names norm and trace have been redefined and unprotected" } {TEXT 280 1 "\n" }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 124 "The warnings tell us that the commands norm and tra ce have a new meaning after the package is loaded. We shall use only t he " }{TEXT 277 4 "norm" }{TEXT 276 203 " command to calculate the mag nitude of a vector. There are several possible definitions of the norm of a vector, each useful in its own right. To obtain the magnitude as we understand it, use the syntax" }}{PARA 208 "" 0 "" {TEXT 276 1 " \+ " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 11 "norm(v1,2);" }} {PARA 211 "" 1 "" {XPPMATH 20 "6#*$\"#H#\"\"\"\"\"#" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 101 "The \"2\" within the norm command stands for the root of order tw o in our definition of the magnitude. " }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 276 209 "Two of the commands contained in the packag e that are important to us are the dot product and the cross product. \+ Let's define another vector v2 and calculate the dot product and the c ross product of v1 and v2." }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> \+ " 0 "" {MPLTEXT 1 273 23 "v2:=vector([1/2,3,-1]);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I#v2G6\"-I'vectorG6$I*protectedGF)I(_syslibGF%6#7%#\" \"\"\"\"#\"\"$!\"\"" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 15 "dotprod(v1,v2);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6 #\"#9" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 17 "crossprod(v1,v2);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#-I'vectorG6$ I*protectedGF&I(_syslibG6\"6#7%\"\"*\"\"!#F+\"\"#" }{TEXT 279 1 " " }} }{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 79 "A ll of the above commands work fine if you enter vectors as lists. For \+ example:" }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 25 "dotprod([2,0,1],[1,1,3]);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6# \"\"&" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 24 "To add properly defined " }{TEXT 281 7 "vect ors" }{TEXT 276 49 ", or to multiply them by scalars use the command " }{TEXT 277 5 "evalm" }{TEXT 276 43 ". Don't use this command when dea ling with " }{TEXT 281 5 "lists" }{TEXT 276 15 ". For example:" }} {PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 21 "eval m(3*v1-(1/2)*v2);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#-I'vectorG6$I*pr otectedGF&I(_syslibG6\"6#7%#\"#B\"\"%#\"#:\"\"##!#BF0" }{TEXT 279 1 " \+ " }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 121 "If you are trying to get away with vectors defined as lists, the \+ syntax for algebraic operations is simpler. For example:" }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 20 "3*[1,2,3]-2*[0 ,1,1];" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#7%\"\"$\"\"%\"\"(" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "" 0 "" {TEXT 276 19 "The useful command " }{TEXT 277 10 "normalize " }{TEXT 276 136 "returns a unit vector wi th the same direction as a given vector. In other words, divides a non zero vector by its magnitude. For example:" }}{PARA 208 "" 0 "" }} {EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 14 "normalize(v1);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#-I'vectorG6$I*protectedGF&I(_syslibG6\"6#7% ,$*$\"#H#\"\"\"\"\"##F0F-,$F,#\"\"$F-,$F,#!\"%F-" }{TEXT 279 1 " " }}} {EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 25 "An other handy command is " }{TEXT 277 5 "angle" }{TEXT 276 55 ". The com mand calculates the angle between two vectors." }}{PARA 208 "" 0 "" {TEXT 276 1 " " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 23 "angle (v1,v2); evalf(%);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#-I'arccosG6$I*p rotectedGF&I(_syslibG6\"6#,$*(\"#H#\"\"\"\"\"#\"#TF-\"\"%F-#\"#9\"%*= \"" }{TEXT 279 1 " " }}{PARA 211 "" 1 "" {XPPMATH 20 "6#$\"+')=!>B'!#5 " }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 46 "The value of the angle is returned in radians." }} {PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 276 27 "The very important \+ command " }{TEXT 277 4 "grad" }{TEXT 276 100 " that calculates the gra dient of a given function of two or more variables is also contained i n the " }{TEXT 277 7 "linalg " }{TEXT 276 29 "package. It works as fol lows:" }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 20 "grad(x^2+y^3,[x,y]);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#-I've ctorG6$I*protectedGF&I(_syslibG6\"6#7$,$I\"xGF(\"\"#,$*$I\"yGF(F-\"\"$ " }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "" 0 "" {TEXT 276 2 "or" }}} {EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 34 "h:=(x,y,z)->exp(2*x)*sin (z*x*y^2);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I\"hG6\"f*6%I\"xGF%I\" yGF%I\"zGF%F%6$I)operatorGF%I&arrowGF%F%*&-I$expGF%6#,$9$\"\"#\"\"\"-I $sinGF%6#*(9&F5F3F59%F4F5F%F%F%" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 30 "gradh:=grad(h(x,y,z),[x,y,z]);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I&gradhG6\"-I'vectorG6$I*protectedGF)I(_sy slibGF%6#7%,&*&-I$expGF(6#,$I\"xGF%\"\"#\"\"\"-I$sinGF(6#*(I\"zGF%F5F3 F5I\"yGF%F4F5F4**F/F5-I$cosGF(F8F5F:F5F;F4F5,$*,F/F5F=F5F:F5F3F5F;F5F4 **F/F5F=F5F3F5F;F4" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" }} {EXCHG {PARA 208 "" 0 "" {TEXT 276 12 "The command " }{TEXT 277 4 "gra d" }{TEXT 276 273 " returns the gradient of a given function as a vect or of the appropriate length in terms of independent variables. To ext ract a component of the gradient vector, use a similar syntax as for l ists. For example, to extract the third component of the gradient, use the command" }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 9 "gradh[3];" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#**-I$e xpG6$I*protectedGF'I(_syslibG6\"6#,$I\"xGF)\"\"#\"\"\"-I$cosGF&6#*(I\" zGF)F.F,F.I\"yGF)F-F.F,F.F4F-" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 6 "Since " }{XPPEDIT 18 0 "gradh = [h[x], h[y], h[z]];" "6#/%&gradhG7%&%\"hG6#%\"xG&F'6#%\"yG& F'6#%\"zG" }{TEXT 282 1 " " }{TEXT 276 105 " , the third component of \+ the gradient is equal to the partial derivative of h with respect to z . Indeed:" }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 17 "diff(h(x,y,z),z);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#**-I$e xpG6$I*protectedGF'I(_syslibG6\"6#,$I\"xGF)\"\"#\"\"\"-I$cosGF&6#*(I\" zGF)F.F,F.I\"yGF)F-F.F,F.F4F-" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 210 "" 0 "" {TEXT 275 39 "Gradients of Function s of Two Variables" }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 " " 0 "" {TEXT 276 74 "You know that for a given function f, say a funct ion of two variables, f= " }{XPPEDIT 18 0 "f(x,y);" "6#-%\"fG6$%\"xG% \"yG" }{TEXT 282 1 " " }{TEXT 276 39 ", its gradient at a given point \+ (a,b), " }{TEXT 283 10 "gradf(a,b)" }{TEXT 276 201 ", is a vector poin ting in the direction of the greatest rate of change of f at (a,b), an d the magnitude of the gradient is equal to that maximum rate of chang e. You also know that for any unit vector " }{XPPEDIT 18 0 "u = (u[1] , u[2]);" "6#/%\"uG6$&F$6#\"\"\"&F$6#\"\"#" }{TEXT 282 1 " " }{TEXT 276 60 ", the directional derivative of f at (a,b) in the direction " }{TEXT 281 2 "u " }{TEXT 276 2 ", " }{TEXT 281 1 " " }{XPPEDIT 18 0 "f [u](a,b);" "6#-&%\"fG6#%\"uG6$%\"aG%\"bG" }{TEXT 282 1 " " }{TEXT 276 55 ", represents the rate of change of f in the direction " }{TEXT 281 1 "u" }{TEXT 276 49 ", and is given as the dot product of the vect or " }{TEXT 281 1 "u" }{TEXT 276 39 " with the gradient gradf(a,b), t hat is:" }}{PARA 208 "" 0 "" }{PARA 213 "" 0 "" {XPPEDIT 18 0 "f[u](a, b) = f[x](a,b)*u[1]+f[y](a,b)*u[2];" "6#/-&%\"fG6#%\"uG6$%\"aG%\"bG,&* &-&F&6#%\"xGF)\"\"\"&F(6#F2F2F2*&-&F&6#%\"yGF)F2&F(6#\"\"#F2F2" } {TEXT 284 1 " " }{TEXT 285 2 "= " }{XPPEDIT 18 0 "gradf(a,b)*u;" "6#*& -%&gradfG6$%\"aG%\"bG\"\"\"%\"uGF)" }{TEXT 284 1 " " }{TEXT 285 2 " ." }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 292 "Because the gradient represents the direction of the greatest rat e of change, gradient vectors gradf(x,y) are perpendicular to the cont ours of f. Calculating gradients, directional derivatives, as well as \+ illustrating the perpendicularity of gradients to contours can be easi ly done in Maple." }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 278 11 "Example 1. " }{TEXT 276 273 " A square flat metal \+ plate, 2 by 2 meters, is positioned on the xy-plane in such a way that the x and y coordinates of points on the plate are between 0 and 2. \+ The plate is heated so that the temperature of each point (x,y) on the plate, in degrees Fahrenheit, is given by" }}{PARA 214 "" 0 "" {XPPEDIT 18 0 "T(x,y) = 10*sin((x-1)*(y-1))+100;" "6#/-%\"TG6$%\"xG%\" yG,&*&\"#5\"\"\"-%$sinG6#*&,&F'F,F,!\"\"F,,&F(F,F,F2F,F,F,\"$+\"F," } {TEXT 286 1 " " }{TEXT 287 2 " ." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 278 3 "(a)" }{TEXT 276 108 " Plot the 3d-graph of the functi on T to have some idea how the temperature distribution in the plate l ooks." }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 35 "We define the function T in Maple. " } }{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 34 "T:= (x,y)->10*sin((x-1)*(y-1))+100;" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I \"TG6\"f*6$I\"xGF%I\"yGF%F%6$I)operatorGF%I&arrowGF%F%,&-I$sinG6$I*pro tectedGF1I(_syslibGF%6#*&,&9$\"\"\"!\"\"F7F7,&9%F7F8F7F7\"#5\"$+\"F7F% F%F%" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 276 276 "Below, we plot the graph T=T(x,y) in the (x,y,T)-space. By rotating the graph, we have found an orientation that gives a good view of the graph. We have locked the orientation in on the command l ine. Still, you should rotate the graph yourself to get a good idea of its shape." }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 61 "plot3d(T(x,y),x=0..2,y=0..2,axes=framed,orientation =[15,80]);" }}{PARA 215 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'- %%GRIDG6%;$\"\"!F($\"\"#F(F&X,I)anythingGI*protectedGF-6\"F.[gl'!%\"!! #\\bm\":\":405B1A8A9B2CB503405AFBE3C98B76DE405AD9B6935C553B405AB43FB1C 6F72D405A8BC1B5628BD4405A60848FF0EF46405A32D5148AEE3B405A03046F20C65D4 059D1679441690840599E56AA28BF6B40596A2C6C2120B340593545894E40254059000 0000000004058CABA76B1BFDB405895D393DEDF4D405861A955D7409540582E986BBE9 6F84057FCFB90DF39A34057CD2AEB7511C540579F7B700F10BA4057743E4A9D742C405 74BC04E3908D3405726496CA3AAC54057041C367489234056E57564D34AFE405AFBE3C 98B76DE405ADCB0B37797A4405ABAB5DE411A25405A962606243068405A6F37C251F0A 3405A4625336DF627405A1B2BACC415B84059EE8B58B7BB724059C086D8F5A97740599 162E2F7238C40596165D96B1337405930D7631E517840590000000000004058CF289CE 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" }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 278 3 "(b)" }{TEXT 276 139 " Plot the co ntour diagram and the field of gradients for the temperature distribut ion T(x,y) in one coordinate system. What do you observe?" }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 276 57 "To plot the contour diagr am, we need to load the package " }{TEXT 277 5 "plots" }{TEXT 276 27 " that contains the command " }{TEXT 277 11 "contourplot" }{TEXT 288 26 ", as well as the commands " }{TEXT 277 9 "fieldplot" }{TEXT 288 5 " and " }{TEXT 277 7 "display" }{TEXT 288 19 " that we use below." }} {PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 12 "with (plots):" }}{PARA 212 "" 1 "" {TEXT 280 49 "Warning, the name changeco ords has been redefined" }{TEXT 280 1 "\n" }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 217 "Now we shall create the c ontour diagram of T(x,y) throughout the plate. We shall label the diag ram c, and end the command with a colon as we don't want the graph dis played at this point. Observe that we use the entry " }{TEXT 277 11 "c ontours=15" }{TEXT 276 11 " under the " }{TEXT 277 12 "contourplot " } {TEXT 276 138 "command. The entry increases the number of contours sho wn. The default setting of 8 contours often gives graphs that are not \+ clear enough." }}{PARA 208 "" 0 "" {TEXT 276 1 " " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 105 "c:=contourplot(T(x,y),x=0..2,y=0..2, contours=15,filled=true,coloring=[white,yellow],scaling=constrained):" }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 166 "The next step is to create a plot of the field of gradients for t he function T(x,y). This can be accomplished in several ways. For exam ple, you could use the command " }{TEXT 277 8 "gradplot" }{TEXT 276 195 " that plots the field of gradients without actually calculuating \+ them. Since we are going to need the explicit form for the gradient of T(x,y) below, we shall calculate the gradient, and use the " }{TEXT 277 9 "fieldplot" }{TEXT 276 55 " command. The command plots any given field of vectors." }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 26 "gradT:=grad(T(x,y),[x,y]);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I&gradTG6\"-I'vectorG6$I*protectedGF)I(_syslibGF%6#7$, $*&-I$cosGF(6#*&,&I\"xGF%\"\"\"!\"\"F5F5,&I\"yGF%F5F6F5F5F5F7F5\"#5,$* &F/F5F3F5F9" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 212 "Maple has calculated the gradient gra dT at any point (x,y). Now we shall create a plot of the field of grad ients and label it g. We do not want to display it at this point, henc e, we end the command with a colon." }}}{EXCHG {PARA 215 "" 1 "" }} {EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 67 "g:=fieldplot(gradT,x=0.. 2,y=0..2,scaling=constrained,arrows=thick):" }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 278 7 "Note. " }{TEXT 283 237 " If you want the field of gradients or the contourplot above displayed \+ separately, replace colons with semicolons, and remove the labels c an d g. 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45.0 45.0 1 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 1 2" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" } }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 491 "What do we observe? The gradients are perpendicular to the contou rs. They point to the direction of increase of the function T(x,y). (R emember that darker shades of yellow correspond to higher temperatures in our example.) We also notice that longer gradient vectors correspo nd to contours being closer together, shorter to contours being farthe r apart. This is not surprising as contours close to each other signif y the fast change in a function and hence a large magnitude of the gra dient." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 278 6 "Note: " } {TEXT 283 82 "In order to see perpendicularity of gradient vectors and contours you have to use " }{TEXT 277 20 "scaling=constrained " } {TEXT 283 7 "option!" }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 278 5 "Note." }{TEXT 276 1 " " }{TEXT 283 382 "The gradient arrows dis played by Maple are scaled in length. The length of each arrow represe nts a relative magnitude of the gradient and not the real magnitude co rresponding to units in a given coordinate system. The scaling is nece ssary in order to obtain a picture that shows the qualitative behavior of the gradient field rather than a bunch of long vectors, scrambled \+ together." }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 278 5 "(c) " }{TEXT 276 194 "Find the temperature at the point \+ p=(.5,.5), that is, at the point x=.5, y=.5. Find the gradient gradT(. 5,.5) and its magnitude at the point p. What do the gradient and its m agnitude represent? " }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 276 184 "We have calculated gradT at any point (x,y) above. All we nee d to do is to substitute our point p into the expression for the gradi ent. Because gradT is a vector, we have to apply the " }{TEXT 277 5 "s ubs " }{TEXT 288 3 "and" }{TEXT 277 6 " evalf" }{TEXT 276 100 " comman ds to each component separately. Since we want our answer to be a vect or, we use the command " }{TEXT 277 6 "vector" }{TEXT 276 35 ". We den ote gradT(.5,.5) as gradTp." }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 86 "gradTp:=vector([evalf(subs(x=0.5,y=0.5,grad T[1])),evalf(subs(x=0.5,y=0.5,gradT[2]))]);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I'gradTpG6\"-I'vectorG6$I*protectedGF)I(_syslibGF%6#7$ $!+3@cW[!\"*F-" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" }} {EXCHG {PARA 208 "" 0 "" {TEXT 276 196 "The vector gradTp shows the di rection from the point p in which the function increases fastest. How \+ fast does the function increase in this direction? This is given by th e magnitute of the vector." }}{PARA 208 "" 0 "" {TEXT 276 2 " " }}} {EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 15 "norm(gradTp,2);" }} {PARA 211 "" 1 "" {XPPMATH 20 "6#$\"+PaC^o!\"*" }{TEXT 279 1 " " }}} {EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 143 "I f you move from the point (.5,.5) in the direction of the vector gradT p, the temperature increases at the instantaneous rate of approximatel y " }{XPPEDIT 18 0 "6.85^o;" "6#)$\"$&o!\"#%\"oG" }{TEXT 282 1 " " } {TEXT 276 39 "F/m. The temperature at the point p is:" }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 11 "T(0.5,0.5);" }} {PARA 211 "" 1 "" {XPPMATH 20 "6#$\"+'RSZ-\"!\"(" }{TEXT 279 1 " " }}} {EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 19 "de grees Fahrenheit." }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 278 4 "(d) " }{TEXT 276 142 "Plot the contour diagram and \+ the field of gradients for the function T(x,y) in a small region aroun d the point p=(.5,.5). What do you observe?" }}{PARA 208 "" 0 "" } {PARA 208 "" 0 "" {TEXT 276 95 "Again, we shall create and label the t wo plots, and then display them in one coordinate system." }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 77 "c1:=contou rplot(T(x,y),x=0.4..0.6,y=0.4..0.6,scaling=constrained,color=blue):" } }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 76 "g1:=fieldplot(gradT,x= 0.4..0.6,y=0.4..0.6,scaling=constrained,arrows=thick):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 17 "display([c1,g1]);" }}{PARA 215 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6'-%'CURVESG6iy7$7$$\"3u*** *********zS!#=$\"3*fb#=wwC/VF*7$$\"3WCDDTEejSF*$\"3^++++++?VF*7$F-7$$ \"3C+++++++SF*$\"36Sr/h))=!Q%F*7$7$$\"3P++++++gTF*$\"3V\\3.\")\\AEUF*7 $$\"3&[P4\">Q.YTF*$\"3V++++++SUF*7$F>7$$\"3I++++++!3%F*$\"3bcD=wwC/VF* 7$7$FAF?7$Fe\\,VF*$\"3k++++++![%F*7$FfpF io7$7$$\"3c+++++++WF*$\"3:\"**4dH.HQ%F*7$F`qF^q7$Fbq7$F0$\"3<=*=oo;?Y% F*7$7$FipFgp7$FdpF07$7$FeqF0F]q7$7$F_pF]p7$FjoFA7$F^rFhq7$7$FeoFco7$F` 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m-%%FONTG6$%*HELVETICAGFacn%+HORIZONTALGFgg`m-%(SCALINGG6#%,CONSTRAINE DGFcg`m" 1 2 2 0 10 1 2 6 1 4 1 1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 336 "We observe that in the small square of the size .2 by .2 around t he point p=(.5,.5), the contours are almost parallel, equally spaced l ines, with gradient vectors clearly perpendicular to them. That is bec ause locally, near p, the fraph of the function T(x,y) is almost plana r; it almost coincides with its tangent plane at that point. " }}} {EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 278 15 "(f ) Heat Flow. " }{TEXT 276 407 "The flow of heat in a temperature field takes place in the direction of maximum decrease of temperature. Phys ical experiments show that the rate of flow v is proportional to the g radient of the temperature, v=-KgradT, where K is a positive constant \+ representing the thermal conductivity of the body. For the plate in E xample 1, plot the field of vectors that qualitatively describes heat \+ flow in the plate." }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 276 247 "The direction of heat flow at any point of the plate (x,y) coinci des with the direction of maximum decrease in temperature, that is, wi th the direction -gradT. The rate is proportional to that vector. Sinc e the lengths of vectors displayed by the " }{TEXT 277 9 "fieldplot" } {TEXT 276 185 " command is scaled anyway, we don't have to worry about the constant K. 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cooler parts." }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 210 "" 0 "" {TEXT 275 41 "Gradients of Functions of Three Variab les" }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 276 507 "The formula s for the gradient and directional derivatives, as well as Maple synta x for calculating them, are very similar as in the case of two variabl es. Visualization of the concepts is, of course, much more difficult. \+ We can plot fields of gradients for functions of three variables and p lot them in the 3d-space together with a couple of level surfaces. How ever, because of our optical limitations, it is only in simple cases t hat we can actually see the perpendicularity of gradients to level sur faces." }}}{EXCHG {PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 278 12 "E xample 2. " }{TEXT 276 64 "Plot a level surface and the field of grad ients for the function" }}{PARA 216 "" 0 "" {XPPEDIT 18 0 "F(x,y,z) = \+ 2*x^2+y^2+z^2;" "6#/-%\"FG6%%\"xG%\"yG%\"zG,(*&\"\"#\"\"\"*$F'F,F-F-*$ F(F,F-*$F)F,F-" }{TEXT 289 1 " " }{TEXT 290 2 " ." }}{PARA 208 "" 0 "" {TEXT 276 1 " " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 26 "F:=( x,y,z)->2*x^2+y^2+z^2;" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I\"FG6\"f* 6%I\"xGF%I\"yGF%I\"zGF%F%6$I)operatorGF%I&arrowGF%F%,(*$9$\"\"#F1*$9%F 1\"\"\"*$9&F1F4F%F%F%" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 30 "gradF:=grad(F(x,y,z),[x,y,z]);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I&gradFG6\"-I'vectorG6$I*protectedGF)I(_syslibGF%6#7% ,$I\"xGF%\"\"%,$I\"yGF%\"\"#,$I\"zGF%F2" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 88 "G:=fieldplot3d(gradF,x=-1..1,y= -1..1,z=-1..1,scaling=constrained,axes=framed,color=red):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 86 "L:=implicitplot3d(F(x,y,z)=1,x= -1..1,y=-1..1,z=-1..1,axes=framed,scaling=constrained):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 15 "display([G,L]);" }}{PARA 215 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6(-%'CURVESG6][m7%7%$!3ASO18j dL))!#=$!36?=`c\")y;%*F*F+7%$!3)fj$poBk;6!#<$!35=oM%=@$e5F0F17%$!3s\"z $QXWb^5F0$!3Hd'3H'zAi5F0$!3C[_tqt1L5F07%7%$!3dGSZmtp4jF*F+F+7%$!3Yd-C> p,wzF*F1F17%$!3OVO\\#>X%puF*$!35E[9lj*Q0\"F0F87%7%$!3Z%=ey$F*F+F+ 7%$!3gaTa^,h&y%F*F1F17%$!3!yO\\6$fMBWF*$!3\"\\*4QnZcX5F0F87%7%$!351[Ht %R>E\"F*F+F+7%$!3+_![QQ._f\"F*F1F17%$!33$40)pmCx8F*$!3sjrhpJBP5F0F87%7 %$\"3F0[Ht%R>E\"F*F+F+7%$\"3g]![QQ._f\"F*F1F17%$\"3k\"=R:f_)o;F*$!3vKL &=d,*G5F0F87%7%$\"3O;W))>%=ey$F*F+F+7%$\"3\\`Ta^,h&y%F*F1F17%$\"3kcM)G &=&\\r%F*$!3c,&*3u*p0-\"F0F87%7%$\"3YFSZmtp4jF*F+F+7%$\"3Nc-C>p,wzF*F1 F17%$\"3kJxA960hxF*$!3PqcKw$QA,\"F0F87%7%$\"36RO18jdL))F*F+F+7%$\"3)fj $poBk;6F0F1F17%$\"3y+sbP]r!3\"F0$!3>R=cyn!R+\"F0F87%7%F($!3z&elYDxis'F *F+7%F.$!3C+([5.P%fvF*F17%$!3@2dEW-sb5F0$!3?0I%)p&e1n(F*F87%7%F)z78=.6^(F*$!3U%p/AfUte(F*F87%7%FHFjqF+7%FKF]rF17%$!3vB& o*>R+lWF*$!3a#QmXhES](F*F87%7%FTFjqF+7%FWF]rF17%$!3v[UieY!*=9F*$!3wr!G pj52U(F*F87%7%FjnFjqF+7%F]oF]rF17%$\"3(f-?Fg%>F;F*$!3()f(*GfYRPtF*F87% 7%FfoFjqF+7%FioF]rF17%$\"3D,V1kQHtYF*$!3)zW^;oySD(F*F87%7%FbpFjqF+7%Fe pF]rF17%$\"3qv&3a7$R>xF*$!3?PJ,/FwqrF*F87%7%F^qFjqF+7%FaqF]rF17%$\"32& Gv'Q#\\l2\"F0$!3JD[PEnW(3(F*F87%7%F($!3Z^$*z_jwNSF*F+7%F.$!3f?#H'=AmNX F*F17%$!3\"HiZJ/'))f5F0$!3xR%*f5v.>ZF*F87%7%F8q 6w_vF*$!3VG6'H`@dj%F*F87%7%FHF^vF+7%FKFavF17%$!39zwy3>m1XF*$!35Y+gtdQ%F*F87%7%FfoF^vF+7%FioFavF17% $\"3JX^CvejJYF*$!3a#)ySAwX-VF*F87%7%FbpF^vF+7%FepFavF17%$\"3'3U*eO^txw F*$!3@r&pZkT\">UF*F87%7%F^qF^vF+7%FaqFavF17%$\"3fpLzRMQs5F0$!3))f78nc# e8%F*F87%7%F($!3rWf(F*$!3*=c$[XF*$!3b]#zg\\%y+;F*F87%7%FTFbzF +7%FWFezF17%$!33gDEO1A-:F*$!3AR4W=&ou^\"F*F87%7%FjnFbzF+7%F]oFezF17%$ \"3k9<3D'yQa\"F*$!3hFE!3a_TV\"F*F87%7%FfoFbzF+7%FioFezF17%$\"3\"**)fU' )y(**e%F*$!3F;V;jl$3N\"F*F87%7%FbpFbzF+7%FepFezF17%$\"3\"\\Eqx9xgj(F*$ !3%\\+Ebe?vE\"F*F87%7%F^qFbzF+7%FaqFezF17%$\"35a9\"4k<#o5F0$!3Y$p()yg/ U=\"F*F87%7%F($\"3g;J$4Xb_M\"F*F+7%F.$\"3FR(4iS()=^\"F*F17%$!35a9\"4k< #o5F0$\"3@#p()yg/U=\"F*F87%7%F$[XF*$\"3W\\#zg\\%y+;F*F87%7%FbpFf^lF+7%FepFi^lF 17%$\"3'*36&*e\">Wf(F*$\"3ygvrt/5%o\"F*F87%7%F^qFf^lF+7%FaqFi^lF17%$\" 3SQ&H?%=0k5F0$\"3RseN^kTn U*eO^txwF*$\"3ap&pZkT\">UF*F87%7%FHFjblF+7%FKF]clF17%$!3UY^CvejJYF*$\" 3)3)ySAwX-VF*F87%7%FTFjblF+7%FWF]clF17%$!3qr3!RhObe\"F*$\"3@#>Y+gtdQ%F *F87%7%FjnFjblF+7%F]oF]clF17%$\"3..MWZEcg9F*$\"3b.Xox&*3pWF*F87%7%FfoF jblF+7%FioF]clF17%$\"3.ywy3>m1XF*$\"3W:GKbbS_XF*F87%7%FbpFjblF+7%FepF] clF17%$\"3.`>8q6w_vF*$\"3xE6'H`@dj%F*F87%7%F^qFjblF+7%FaqF]clF17%$\"3 \"HiZJ/'))f5F0$\"35Q%*f5v.>ZF*F87%7%F($\"3o%elYDxis'F*F+7%F.$\"38*p[5. 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Can you see the perpendicularity of the gradients to the level sur face? " }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 276 135 "The grad ient of a function of three variables has the same interpretation in t erms of rates of change as in the case of two variables. " }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 278 12 "Example 3 . " }{TEXT 276 383 "Suppose that a large rectangular tank of water oc cupies the region 0 " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 69 " We first define the function C in Maple and then obtain its gradient." }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 52 "C :=(x,y,z)->10*exp(-(sqrt((x-2)^2+(y-3)^2+z^2)/10));" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I\"CG6\"f*6%I\"xGF%I\"yGF%I\"zGF%F%6$I)operatorGF%I &arrowGF%F%,$-I$expG6$I*protectedGF2I(_syslibGF%6#,$-I%sqrtGF16#,(*$,& 9$\"\"\"!\"#F=\"\"#F=*$,&9%F=!\"$F=F?F=*$9&F?F=#!\"\"\"#5FHF%F%F%" } {TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 30 "gradC :=grad(C(x,y,z),[x,y,z]);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I&gradC G6\"-I'vectorG6$I*protectedGF)I(_syslibGF%6#7%,$*(,.*$I\"xGF%\"\"#\"\" \"F1!\"%\"#8F3*$I\"yGF%F2F3F7!\"'*$I\"zGF%F2F3#!\"\"F2,&F1F2F4F3F3-I$e xpGF(6#,$*$F/#F3F2#F<\"#5F3F;,$*(F/F;,&F7F2F8F3F3F>F3F;,$*(F/F;F:F3F>F 3F<" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 267 "The formula for the gradient isn't very inv iting. All we need to do, however, is to evaluate the gradient at the \+ point p= (0,5,1). Let's label the result gradCp. Since we have three c oordinates into which we have to substitute x=0, y=5, z=1, we shall us e the command " }{TEXT 277 3 "seq" }{TEXT 276 1 "." }}{PARA 208 "" 0 " " }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 64 "gradCp:=vector([seq( evalf(subs(x=0,y=5,z=1,gradC[i])),i=1..3)]);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#>I'gradCpG6\"-I'vectorG6$I*protectedGF)I(_syslibGF%6#7% $\"+P\")yQ\\!#5$!+P\")yQ\\F/$!+pSRpCF/" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 381 "The vect or gradCp gives the direction from (0,5,1) in which the rate of increa se of the concentration of salt is largest. Physically, we would expec t the direction to be toward the lump of salt. Is it so? The coordinat es of the lump are (2,3,0). (Remember, the salt is located in the midd le of the bottom of the tank.) Let's look at the displacement vector f rom (0,5,1) to (2,3,0)." }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 27 "d:=vector([2,3,0]-[0,5,1]);" }}{PARA 211 "" 1 " " {XPPMATH 20 "6#>I\"dG6\"-I'vectorG6$I*protectedGF)I(_syslibGF%6#7%\" \"#!\"#!\"\"" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 91 "Are gradCp and d parallel? One way to \+ check is to calculate the angle between gradCp and d." }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 26 "angle(gradCp,d); \+ evalf(%);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#-I'arccosG6$I*protectedG F&I(_syslibG6\"6#,$*$\"\"*#\"\"\"\"\"#$\"+MLLLL!#5" }{TEXT 279 1 " " } }{PARA 211 "" 1 "" {XPPMATH 20 "6#$\"\"!F$" }{TEXT 279 1 " " }}} {EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 230 "T hey are, indeed, parallel. Thus, the concentration increases fastest w hen we move in the direction of the lump of salt. What is this maximal rate of increase in concentration? This is given, of course, by the m agnitude of gradCp." }}{PARA 208 "" 0 "" {TEXT 276 1 " " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 15 "norm(gradCp,2);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#$\"+1A=3u!#5" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 84 "The concentrati on is increasing at the rate of, approximately, .741 g/l per meter. " }}}{EXCHG {PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 278 4 "(b) " } {TEXT 276 111 "Find the rate of change in the concentration if we begi n to move from (0,5,1) in the direction toward (2,6,1). " }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 276 272 "Note that the movement is in the direction of the displacement vector from (0,5,1) to (2,6,1), tha t is, [2,1,0]. We calculate the corresponding unit vector u in the dir ection of [2,1,0]. The dot product of u with the gradient gradCp gives us the desired rate of change. " }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 22 "u:=normalize([2,1,0]);" }} {PARA 211 "" 1 "" {XPPMATH 20 "6#>I\"uG6\"-I'vectorG6$I*protectedGF)I( _syslibGF%6#7%,$*$\"\"&#\"\"\"\"\"##F2F/,$F.#F1F/\"\"!" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 273 28 "dotprod(gradCp,u); evalf(%);" }}{PARA 211 "" 1 "" {XPPMATH 20 "6#,$*$\"\"&#\"\"\"\"\"#$ \"+widx)*!#6" }{TEXT 279 1 " " }}{PARA 211 "" 1 "" {XPPMATH 20 "6#$\"+ +Kp3A!#5" }{TEXT 279 1 " " }}}{EXCHG {PARA 208 "> " 0 "" }{PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 276 140 "When we move in the \+ direction [2,1,0] from the point (0,5,1), the concentration increases \+ at the rate of, approximately, .221 g/l per meter." }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 210 "" 0 "" {TEXT 275 17 "Homework Proble ms" }}{PARA 208 "" 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 278 11 "Probl em 1. " }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 278 5 "(a) " } {TEXT 276 22 "Consider the function " }{XPPEDIT 18 0 "f(x,y) = 5/(x^2+ y^2+1);" "6#/-%\"fG6$%\"xG%\"yG*&\"\"&\"\"\",(*$F'\"\"#F+*$F(F.F+F+F+! \"\"" }{TEXT 282 1 " " }{TEXT 276 210 " . Display both its contour diagram and its field of gradients on the same plot. Comment on the d irection in which you should move from a point (x,y) to obtain the max imum and minimum rates of change of f. " }}{PARA 208 "" 0 "" }{PARA 208 "" 0 "" {TEXT 278 5 "(b) " }{TEXT 276 82 "Looking back at part (a ) and remembering graphs and contour diagrams of functions " } {XPPEDIT 18 0 "z = x^2+y^2;" "6#/%\"zG,&*$%\"xG\"\"#\"\"\"*$%\"yGF(F)" }{TEXT 282 1 " " }{TEXT 276 5 " and " }{XPPEDIT 18 0 "z = exp(-(x^2+y ^2));" "6#/%\"zG-%$expG6#,$,&*$%\"xG\"\"#\"\"\"*$%\"yGF,F-!\"\"" } {TEXT 282 1 " " }{TEXT 276 121 " that you have seen before, what can y ou say about contour diagrams and gradient fields of functions of the \+ special form " }{XPPEDIT 18 0 "f(x,y) = g(r);" "6#/-%\"fG6$%\"xG%\"yG- %\"gG6#%\"rG" }{TEXT 282 1 " " }{TEXT 276 8 ", where " }{XPPEDIT 18 0 "r = sqrt(x^2+y^2);" "6#/%\"rG-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F ," }{TEXT 282 1 " " }{TEXT 276 28 " ? What characteristics of " } {XPPEDIT 18 0 "g(r);" "6#-%\"gG6#%\"rG" }{TEXT 282 1 " " }{TEXT 276 107 " will determine the directions that give maximum and minimum rate s of change and magnitudes of those rates?" }}}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "> " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 278 12 "Problem 2. " }{TEXT 276 245 "A room occupies the region 0 " 0 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 278 12 "Problem 3. " }{TEXT 276 439 "In a room that occupies the region 0 " 0 "" }} {EXCHG {PARA 208 "" 0 "" {TEXT 281 97 "This MTH 243 Maple Worksheet wr itten by D. Abrahamson, B. Kaskosz, and L. Pakula, Copyright 1999." }} }{EXCHG {PARA 208 "> " 0 "" }}{PARA 220 "" 0 "" }}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }