Some Skills for Math 272 in Maple You should start with the New Users Tour. Run through the first six sections of the tour. At any time, you can also get help from the help menu.

It is wise to restart. restart: These packages contain some extra routines that may be useful for the sections below. with(linalg): with(plottools): with(plots):

It is easy to plot a function of 2 variables. The following plots f(x,y) = x^2 - y^2 over the region [-2..2]x[-2..2]. plot3d(x^2 - y^2,x=-2..2,y=-2..2); You can add options to make the picture easier to view. In the following a box around the graph has been added and the plot is a wireframe plot rather than a patch plot. Use the help for plot3d[option] to experiment with other possibilities. plot3d(x^2 - y^2,x=-2..2,y=-2..2,axes=boxed, style=wireframe);

Plotting arbitrary surfaces is harder. There is a routine in the plots package called implicitplot3d that can be used to plot surfaces. It must be loaded before one can use it. with(plots,implicitplot3d); This is how one can plot a quadric surface. implicitplot3d(x^2/2 + y^2 - z^2 =0,x=-6..6,y=-6..6, z=-6..6,style=patch,axes=normal);

To plot in cylindrical coordinates where z is a function of r and theta one can use plot3d for a parametric surface. Specifying the cylindrical coordinate system is done as an option. For example, to plot z = r^2 one could use the following. plot3d([r,theta,r^2],r=0..2,theta=0..2*Pi, coords=cylindrical,axes=normal,style=patch, scaling=constrained); In Maple r is assume to be a function of z and theta when using cylindrical coordinates . Notice the order of theta and z. What follows is a plot of 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. Try changing the order of theta and z to see what happens. plot3d(z,theta=0..2*Pi,z=-2..2, coords=cylindrical,axes=normal,style=patch);

One can also plot in spherical coordinates. Here rho is a function of theta and phi. Again the order of the ranges is important. For more information see plot3d,coords. plot3d(sin(phi),theta=0..2*Pi,phi=0..Pi, axes=boxed,style=patch,coords=spherical);

In order to use some of the linear algebra routines one must load them from the linalg package. It is often easier to load the whole package. with(linalg):

One defines a vector using the vector command. The input for this command is a one dimensional array. For example, to input the vectors (1,-1,0) and (2,-3,2) one uses the following commands. a := vector([1,-1,0]); b := vector([2,-3,2]); The way to specify indices for components is with square brackets. For example, the second component of b is obtained by the following. b[2];

The matadd command loaded in the linalg package can be used to add vectors. matadd(a,b); One can do this in several other ways. Here are two. Note the use of the evalm command to evaluate a matrix operation. evalm(a+b); evalm(a &+ b); One can find the magnitude of a vector using the norm command. Since there are other length definitions, you must tell Maple which norm to use. That is why there is the 2 in the following command. norm(a,2);

There are commands for the cross and dot products. The commands are crossprod and innerprod. Here are the dot and cross products for the vectors a and b. innerprod(a,b); crossprod(a,b); crossprod(b,a);

First one needs a function. This is a definition as a function of a vector. f := x -> exp(-abs(x[1]^3-x[1])- x[2]^2); Here is a plot of the function. It has an interesting shape. plot3d(f([x,y]),x=-2..2,y=-2..2, axes=boxed,style=patch);

There are several ways to get contours. The plots package has two specialized routines, contourplot and contourplot3d. One can also use an option in the plot3d function to get contours. This is an example of the contourplot routine. Note that it gives a two dimensional plot. contourplot(f([x,y]),x=-2..2,y=-2..2); This is an example of the contourplot3d. It gives a 3 dimensional plot. contourplot3d(f([x,y]),x=-2..2,y=-2..2, axes=boxed); Finally is the plot3d command. One only needs to use one of the options style=patchcontour or style=contour to get a contour plot. plot3d(f([x,y]),x=-2..2,y=-2..2, style=patchcontour,axes=boxed);

Maple has plotting routines that allow one to plot vector fields. These work in two and three dimensions For two dimensional vector fields one uses fieldplot. Here is a plot of the vector field 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 . fieldplot([x,y],x=-2..2,y=-2..2); This is how one plots a three dimensional vector field using the command fieldplot3d. The function is 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 . fieldplot3d([x,y,z],x=-2..2,y=-2..2, z=-2..2,axes=boxed);

Maple will try to take limits of multivariate functions. Unfortunately, the routine is not very robust. Here are several examples and limits from the text. If Maple does not get an answer, it echos the input. f := x^2 + y^2; limit(f,{x=1,y=-2}); Example 3.2.3 f := x^4/(x^2+y^2); limit(f,{x=0,y=0}); Example 3.2.5 f := (x^2-y^2)/(x^2+y^2); limit(f,{x=0,y=0}); Example 3.2.6 f := (x+y)^2/(x-y); limit(f,{x=1,y=1}); limit(f,{x=0,y=0}); Example 3.2.7 f := x^2*y/(x^4+y^2); limit(f,{x=0,y=0});

The idea of a partial derivative is to view a function of several variables, say 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, as a function of only one of the variable. If we take the variable to be 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, then our function is 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. The following shows how this works. Here 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 is a function of two variables and NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiMtSSNtaUdGJTY5USJnRigvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GKC8lJXNpemVHUSMxMkYoLyUlYm9sZEdRJmZhbHNlRigvJSdpdGFsaWNHUSV0cnVlRigvJSp1bmRlcmxpbmVHRjgvJSpzdWJzY3JpcHRHRjgvJSxzdXBlcnNjcmlwdEdGOC8lK2ZvcmVncm91bmRHUShbMCwwLDBdRigvJStiYWNrZ3JvdW5kR0ZELyUnb3BhcXVlR0Y4LyUrZXhlY3V0YWJsZUdGOC8lKXJlYWRvbmx5R0Y4LyUpY29tcG9zZWRHRjgvJSpjb252ZXJ0ZWRHRjgvJStpbXNlbGVjdGVkR0Y4LyUscGxhY2Vob2xkZXJHRjgvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RigvJSptYXRoY29sb3JHRkQvJS9tYXRoYmFja2dyb3VuZEdGRC8lK2ZvbnRmYW1pbHlHRjIvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YoLyUpbWF0aHNpemVHRjU3IzYjJSJnRw== is the trace aprallel to the y-axis through 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. f := x -> x[1]^3 + x[2]^2; g := y -> f([1,y]); eval(g(1)); One can plot the function 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 and the function NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZELyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1GJDYmLUYtNjlRImdGKEYwRjNGNkY5RjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRmhuRltvLUkjbW9HRiU2M1EwJkFwcGx5RnVuY3Rpb247RigvJSVmb3JtR1EmaW5maXhGKC8lJmZlbmNlR0Y4LyUqc2VwYXJhdG9yR0Y4LyUnbHNwYWNlR1EkMGVtRigvJSdyc3BhY2VHRl9wLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lMGZvbnRfc3R5bGVfbmFtZUdGVy8lJXNpemVHRjUvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGRC1GJDYlLUZjbzYzUSIoRigvRmdvUSdwcmVmaXhGKC9Gam9GO0ZbcC9GXnBRLnRoaW5tYXRoc3BhY2VGKC9GYXBGY3IvRmNwRjtGZHBGZnBGaXBGXHFGXnFGYHFGYnFGZHFGZnFGaHEtRiQ2Iy1GLTY5USJ5RihGMEYzRjZGOUY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZobkZbby1GY282M1EiKUYoL0Znb1EocG9zdGZpeEYoRmFyRltwRmJyL0ZhcFEydmVyeXRoaW5tYXRoc3BhY2VGKEZlckZkcEZmcEZpcEZccUZecUZgcUZicUZkcUZmcUZocUYsRiw3IzYjLSUiZ0c2IyUieUc= on the same graph to see what is happening. Here 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 is plotted over the square 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 and NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZELyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1GJDYmLUYtNjlRImdGKEYwRjNGNkY5RjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRmhuRltvLUkjbW9HRiU2M1EwJkFwcGx5RnVuY3Rpb247RigvJSVmb3JtR1EmaW5maXhGKC8lJmZlbmNlR0Y4LyUqc2VwYXJhdG9yR0Y4LyUnbHNwYWNlR1EkMGVtRigvJSdyc3BhY2VHRl9wLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lMGZvbnRfc3R5bGVfbmFtZUdGVy8lJXNpemVHRjUvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGRC1GJDYlLUZjbzYzUSIoRigvRmdvUSdwcmVmaXhGKC9Gam9GO0ZbcC9GXnBRLnRoaW5tYXRoc3BhY2VGKC9GYXBGY3IvRmNwRjtGZHBGZnBGaXBGXHFGXnFGYHFGYnFGZHFGZnFGaHEtRiQ2Iy1GLTY5USJ5RihGMEYzRjZGOUY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZobkZbby1GY282M1EiKUYoL0Znb1EocG9zdGZpeEYoRmFyRltwRmJyL0ZhcFEydmVyeXRoaW5tYXRoc3BhY2VGKEZlckZkcEZmcEZpcEZccUZecUZgcUZicUZkcUZmcUZocUYsRiw3IzYjLSUiZ0c2IyUieUc= is plotted along the line 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 across the same square. plot1 := plot3d(f([x,y]),x=0..2,y=0..2, style=patchnogrid,axes=boxed, shading=zgreyscale): plot2 := spacecurve([1,1+t,g(1+t)],t=-1..1,color=red): display(plot1,plot2,orientation=[150,50]); The partial derivative of NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZELyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1GJDYmLUYtNjlRImZGKEYwRjNGNkY5RjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRmhuRltvLUkjbW9HRiU2M1EwJkFwcGx5RnVuY3Rpb247RigvJSVmb3JtR1EmaW5maXhGKC8lJmZlbmNlR0Y4LyUqc2VwYXJhdG9yR0Y4LyUnbHNwYWNlR1EkMGVtRigvJSdyc3BhY2VHRl9wLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lMGZvbnRfc3R5bGVfbmFtZUdGVy8lJXNpemVHRjUvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGRC1GJDYlLUZjbzYzUSIoRigvRmdvUSdwcmVmaXhGKC9Gam9GO0ZbcC9GXnBRLnRoaW5tYXRoc3BhY2VGKC9GYXBGY3IvRmNwRjtGZHBGZnBGaXBGXHFGXnFGYHFGYnFGZHFGZnFGaHEtRiQ2Iy1GLTY5USJ4RihGMEYzRjZGOUY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZobkZbby1GY282M1EiKUYoL0Znb1EocG9zdGZpeEYoRmFyRltwRmJyL0ZhcFEydmVyeXRoaW5tYXRoc3BhY2VGKEZlckZkcEZmcEZpcEZccUZecUZgcUZicUZkcUZmcUZocUYsRiw3IzYjLSUiZkc2IyUieEc= in the 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 direction is the derivative of 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. Here is the derivative of 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 and a 2D plot of NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiMtSSNtaUdGJTY5USJnRigvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GKC8lJXNpemVHUSMxMkYoLyUlYm9sZEdRJmZhbHNlRigvJSdpdGFsaWNHUSV0cnVlRigvJSp1bmRlcmxpbmVHRjgvJSpzdWJzY3JpcHRHRjgvJSxzdXBlcnNjcmlwdEdGOC8lK2ZvcmVncm91bmRHUShbMCwwLDBdRigvJStiYWNrZ3JvdW5kR0ZELyUnb3BhcXVlR0Y4LyUrZXhlY3V0YWJsZUdGOC8lKXJlYWRvbmx5R0Y4LyUpY29tcG9zZWRHRjgvJSpjb252ZXJ0ZWRHRjgvJStpbXNlbGVjdGVkR0Y4LyUscGxhY2Vob2xkZXJHRjgvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RigvJSptYXRoY29sb3JHRkQvJS9tYXRoYmFja2dyb3VuZEdGRC8lK2ZvbnRmYW1pbHlHRjIvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YoLyUpbWF0aHNpemVHRjU3IzYjJSJnRw== and NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiMtSSNtaUdGJTY5USNEZ0YoLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRigvJSVzaXplR1EjMTJGKC8lJWJvbGRHUSZmYWxzZUYoLyUnaXRhbGljR1EldHJ1ZUYoLyUqdW5kZXJsaW5lR0Y4LyUqc3Vic2NyaXB0R0Y4LyUsc3VwZXJzY3JpcHRHRjgvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYoLyUrYmFja2dyb3VuZEdGRC8lJ29wYXF1ZUdGOC8lK2V4ZWN1dGFibGVHRjgvJSlyZWFkb25seUdGOC8lKWNvbXBvc2VkR0Y4LyUqY29udmVydGVkR0Y4LyUraW1zZWxlY3RlZEdGOC8lLHBsYWNlaG9sZGVyR0Y4LyUwZm9udF9zdHlsZV9uYW1lR1ErMkR+Q29tbWVudEYoLyUqbWF0aGNvbG9yR0ZELyUvbWF0aGJhY2tncm91bmRHRkQvJStmb250ZmFtaWx5R0YyLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGKC8lKW1hdGhzaXplR0Y1NyM2IyUjRGdH. Dg := unapply(diff(g(y),y),y); slope_y := Dg(1); plot({g(y),g(1)+(y-1)*slope_y},y=0..2, color=black); Here is a plot of NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZELyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1GJDYmLUYtNjlRImZGKEYwRjNGNkY5RjxGPkZARkJGRUZHRklGS0ZNRk9GUUZTRlVGWEZaRmZuRmhuRltvLUkjbW9HRiU2M1EwJkFwcGx5RnVuY3Rpb247RigvJSVmb3JtR1EmaW5maXhGKC8lJmZlbmNlR0Y4LyUqc2VwYXJhdG9yR0Y4LyUnbHNwYWNlR1EkMGVtRigvJSdyc3BhY2VHRl9wLyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShtYXhzaXplR1EpaW5maW5pdHlGKC8lKG1pbnNpemVHUSIxRigvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lMGZvbnRfc3R5bGVfbmFtZUdGVy8lJXNpemVHRjUvJStmb3JlZ3JvdW5kR0ZELyUrYmFja2dyb3VuZEdGRC1GJDYlLUZjbzYzUSIoRigvRmdvUSdwcmVmaXhGKC9Gam9GO0ZbcC9GXnBRLnRoaW5tYXRoc3BhY2VGKC9GYXBGY3IvRmNwRjtGZHBGZnBGaXBGXHFGXnFGYHFGYnFGZHFGZnFGaHEtRiQ2JS1GLTY5USJ4RihGMEYzRjZGOUY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZobkZbby1GY282M1EiLEYoRmZvRmlvL0ZccEY7Rl1wL0ZhcFEzdmVyeXRoaWNrbWF0aHNwYWNlRihGYnBGZHBGZnBGaXBGXHFGXnFGYHFGYnFGZHFGZnFGaHEtRi02OVEieUYoRjBGM0Y2RjlGPEY+RkBGQkZFRkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm5GaG5GW28tRmNvNjNRIilGKC9GZ29RKHBvc3RmaXhGKEZhckZbcEZici9GYXBRMnZlcnl0aGlubWF0aHNwYWNlRihGZXJGZHBGZnBGaXBGXHFGXnFGYHFGYnFGZHFGZnFGaHFGLEYsNyM2Iy0lImZHNiQlInhHJSJ5Rw==, 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 and the tangent line to the graph of 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 together. (Rotate the graph to get a better idea of what is happening.) plot3 := spacecurve([1,t+1,g(1)+t*slope_y], t=-1..1,color=blue): display(plot1,plot2,plot3,orientation=[15,70]); Now we can also plot the the trace for 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 and the line tangent to the trace at 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. h := x -> f([x,1]); Dh := unapply(diff(h(x),x),x); slope_x := Dh(1); plot({h(x),h(1)+(x-1)*slope_x},x=0..2, color=black); plot4 := spacecurve([t+1,1,h(1+t)],t=-1..1,color=red): plot5 := spacecurve([1+t,1,h(1)+t*slope_x], t=-1..1,color=blue): display({plot1,plot2,plot3,plot4,plot5}, orientation=[40,95]);

If one only wants to get the partial derivative of an expression in Maple one uses the diff command as in the following. expr := x^2*y^2 - sin(2*x-3*y+z); diff(expr,x); diff(expr,x,x); diff(expr,x,y,z);

This shows how one can define a tangent plane with the function 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 from above. One can now use the point 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 and the vectors 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 and 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 to define a plane. Here is a plot of the plane combined with the above plots. tplane := plot3d(evalm([1,1,f([1,1])]+ (t*[1,0,slope_x])+(s*[0,1,slope_y])), s=-1..1,t=-1..1,color=turquoise, style=patchnogrid): display(plot1,plot2,plot3,plot4,plot5,tplane,orientation=[75,65]); The above graph shows how the linear function 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 approximates 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 in the same way that a linear function of one variable is used to approximate a function of one variable. Using this idea, one defines the derivative of a function 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 from 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 to 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 at 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 as a linear transformation 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 such that 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. Here is the limit for the function from above. limit((f([x+1,y+1])-f([1,1])- dotprod([x,y],[slope_x,slope_y]))/ norm([x,y],2),{x=0,y=0}); The limit is actually zero. (Check this!)

This is a procedure to calculate the gradient of a function as a function. It takes two parameters; the function and the dimension of the input vector. fgrad := proc(f,n) local i,x; unapply([seq(eval( diff(f(x),x[i])),i=1..n)],x); end; With this one can easily calculate the the gradient. f := x -> x[1]^2 -4*x[2]^2; gradf := fgrad(f,2); g:= x -> x[1]*cos(x[2])+x[2]*cos(x[1]*x[3]); gradg := fgrad(g,3); Combining the fgrad function with the innerprod function one can calculate directional deivatives. Here is how one could calculate the derivative of g from above in the same direction as [1,-2,1] as a function of the point x. One can easily calculate this at the point (0,1,-1). Du_g := unapply(innerprod(gradg(x),[1,-2,1])/ norm([1,-2,1],2),x); Du_g([0,1,-1]);

With the following routines one can calculate the divergence and curl of a vector field. The fdiv procedure takes in the vector field function and the dimension of the input. The fcurl only takes in the vector field since the dimension is always assumed to be 3. fdiv:= proc(f,n::integer) local i, x; unapply(add(diff(f(x)[i], x[i]), i = 1 .. n), x); end: fcurl := proc(f) local x; unapply(eval([diff(f(x)[3],x[2])-diff(f(x)[2],x[3]), diff(f(x)[1],x[3])-diff(f(x)[3],x[1]), diff(f(x)[2],x[1])-diff(f(x)[1],x[2])]) ,x): end: The following is a vector field, its divergence, and its curl. f := x -> [x[2]*x[1],x[1]^2,x[3]]; divf := fdiv(f,3); curlf := fcurl(f); Here is a plot of the vector field f and the curl at the point (2,2,2). pt := [2,2,2]: fie := fieldplot3d(f([x,y,z]),x=0..4,y=0..4, z=0..4,axes=boxed,color=black,grid=[5,5,5]): cf := arrow(pt,.5*curlf(pt), [1,0,0],.02, .08, .1,color=red): display(fie,cf);

One can plot a parametric curve using the plot routine. Here is a plot of the curve 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. plot([t,t+cos(t),t=-3..3],x=-4..4,y=-4..4, color=violet);

Plotting curves in NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZELyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1GJDYlRiwtSSVtc3VwR0YlNiUtRi02OVEiUkYoRjBGM0Y2RjlGPEY+RkBGQkZFRkdGSUZLRk1GT0ZRRlNGVUZYRlpGZm5GaG5GW28tSSNtbkdGJTY5USIzRihGMEYzRjYvRjpGOEY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbi9GaW5RJ25vcm1hbEYoRltvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGKEYsRiw3IzYjKiQpJSJSRyIiJCIiIg== is simple in Maple. The command that one uses is spacecurve. It is in the plots package. First load the plots package. restart; with(plots):

It is difficult to define a function from 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 to a vector. It is easy if one uses a list. Here is the function 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. f := t -> [t,t^3,sin(t)]; f(2); It is easy to take the derivative using the diff command. The diff command only takes partial derivatives. Note how the unapply command turns the derivative into a function. Df := unapply(diff(f(t),t),t); Df(2);

Now we can plot f(t) and f'(t) using spacecurve. The options for spacecurve are almost the same as for plot3d. spacecurve(f(t),t=-2..2,axes=boxed,color=red); Now we plot f'(t). spacecurve(Df(t),t=-2..2,axes=boxed,color=blue);

There are two different ways to enter a matrix. One can use the palettes or use the matrix command. There are different ways to use the matrix command.

When using the matrix command one can either put in the matrix as a list of lists or one can give the size of the matrix and a list of the elements going across the rows first. B := matrix([[1,2,3],[4,5,6]]); C := matrix(2,3,[7,8,9,10,11,12]);

In order to use the matrix palette one should change the input display style. Go into the options, input display menu. Select the standard math option. Place the curser at the end of this paragraph. Now go to the insert, execution group menu and select the after curser option. Now go to the view, palettes menu and select the matrix palette option. Put the curser at the prompt and setect the 2 by 3 matrix. Fill in the entries from B above. 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

First return the input display style to standard Maple. There are only a few basic matrix operations: addition, multiplication, and scalar multiplication. There are a number of different ways to each of these operations in Maple.

There are three basic ways of adding matrices. They are simple addition, using &+, and using matadd. B+C; C&+B; matadd(C,B); One must use evalm to see the results of the first two. evalm(B+C); evalm(C&+B);

Here there are are two ways to do this. They are similar to two of the way to do matrix addition. evalm(2*B); scalarmul(B,2);

There are two ways to evaluate matrix multiplication. They are related to the operations for addition. We will need a new matrix to match sizes. A := matrix([[1,2],[4,-3]]); multiply(A,B); evalm(A&*B);

The following is how one gets the matrix representation of the transformation 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. One follows a general pattern. Here is an arbitrary transformation from 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 to 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. The matrix is a 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. 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 is the matrix. Multiplying by a two vector gives the coefficient positions. A := matrix([[a,b],[c,d]]); multiply(A,[x[1],x[2]]); This means that one has the coefficients 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, 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, 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, and 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. Hence the transformation corresponds to the matrix 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. Here is a check. A := matrix([[2,1],[1,2]]); multiply(A,[x[1],x[2]]);

The general area change can be infered from what happens to a standard unit square with vertices [0,0], [0,1], [1,1], and [1,0]. Since a linear transformation takes parallel lines to parallel lines one can simply find the area of the new polygon using the cross product, crossprod. Here 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 is applied to 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 and 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 to get the two new vectors. Find the cross product of the vectors in 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 and compare it with det(A). Then compare your results with Theorems 2.4.1 and 2.4.2. multiply(A,[1,0]); multiply(A,[0,1]); det(A);

a linear transformation. This is how one plots a polygon. This is the standard unit square. vertices1 := [[0,0],[1,0],[1,1],[0,1]]; polygonplot(vertices1,view=[-2..2,-2..2],color=green); One can use the map function to use 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 to transform the vertices. This enables one to plot the image of the unit square under 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. Tsquare := map(multiply,vertices1,A); polygonplot(Tsquare,view=[-1..4,-1..4],color=blue);

The following function produces an animation of a linear transformation. You do not need to know how it works. You should know that there are other ways of doing this that may give different animations.

LinTrans2d := proc(A::{matrix,listlist}) local T,t1,p1,listP,B,color1,list1,m; T := (A,t) -> linalg[matadd](linalg[scalarmul](A,t), linalg[scalarmul]([[1,0],[0,1]],(1-t))); list1 := [[0,0],[1,0],[1,1],[0,1]]; p1 := plottools[polygon](list1,color=blue, linestyle=3,thickness=2): m := 1.1*max(1,op(map(abs, [linalg[eigenvals](A)]))); listP := [p1]: t1 :=0; for t1 from 0 by 1/10 to 1 do B := evalm(T(A,t1)); if det(B) >= 0 then color1 := `blue`; else color1 := `red`; fi; listP := [op(listP),plottools[polygon]( map(convert,(map(linalg[multiply], list1,B)),list),color=color1)]; od: print(plots[display](listP, insequence=true,view=[-m..m,-m..m])); end:

Here is the procedure applied to 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. To see the animation, select the graph and then use the new tool bar to start the animation. LinTrans2d(A);

There is a package of routines for calculating with functions of vectors. They are in the following execution group. We do not need all of them for this worksheet. fgrad := proc(f,n) # # This routine takes a real valued function # of a n dimensional vector and returns its # gradient as a function of an n vector. # # The inputs are the function f and n # the dimension of the vector. # # local i,x; x := vector(n); unapply([seq(eval( diff(f(x),x[i])),i=1..n)],x); end: fcompose := proc(G,F) # # This routine takes a pair of vector to # vector functions, composes them and returns # their composition as a vector to vector function. # # The inputs are the functions F and G. # # local x; unapply(G(F(x)),x); end: lmultiply := proc(G,H) # # This routine will take two matrices or listlists, # multiply them as matrices and return a listlist. # This is useful for makeing functions of vectors. # # local ans,G1; ans := linalg[multiply](eval(G),eval(H)); if type(ans,vector) then convert(ans,list); elif type(ans,matrix) then convert(ans,listlist); else ans; fi; end: Dcompose := proc(DG,DF,F) # # This routine takes in the derivative functions # DG and DF and the function F and returns the # derivative D(G_of_F) as a function. # # local x; unapply(lmultiply(DG(F(x)),DF(x)),x) end: fjacobian := proc(f,n) # # This routine takes in a function f from # Rn to Rm and the dimension n. It returns # the Jacobian of f as a function of x. # # local i,j,m,x; x := vector(n); m := linalg[vectdim](f(x)); unapply([seq([seq(diff(f(x)[i],x[j]),j=1..n)],i=1..m)],x); end: fhessian := proc(f,n) # # This routine takes in a function f from Rn to R and the # dimension n and returns the hessian of f as a function # of the vector x. # # local i,j,x; x := vector(n); unapply([seq([seq(diff(f(x),x[i],x[j]),j=1..n)],i=1..n)],x); end: fdiv:= proc(f,n) local i, x; # # This routine takes in a vector function of a vector x and the # dimension of x and return the divergence of f as a # function of the vector x. # # x := vector(n); unapply(add(diff(f(x)[i], x[i]), i = 1 .. n), x); end: fcurl := proc(f) # # This routine takes in a vector function of a vector x and the # dimension of x and return the curl of f as a # function of the vector x. # # local x; unapply(eval([diff(f(x)[3],x[2])-diff(f(x)[2],x[3]), diff(f(x)[1],x[3])-diff(f(x)[3],x[1]), diff(f(x)[2],x[1])-diff(f(x)[1],x[2])]) ,x): end: fD := proc() # # This function takes a function from R^n (as a vector) # to R and a sequence of integers, i1, i2, i3, ..., im # and returns the partial of the function with respect # to i1, i2, i3, ..., im as a function of a vector. # # local f,i,j,k; if nargs =1 then D(args[1]) elif nargs > 1 then f := args[1](x); for i from 2 to nargs do f := eval(diff(f,x[args[i]])); od; unapply(f,x); fi; end: vD := proc() # # This procedure takes a function from R to R^n # and returns the derivative of the function as # a function from R to R^n. # # local f,i,j,k; if nargs =1 then unapply(diff(args[1](t),t),t); elif nargs > 1 then f := eval(args[1](t)); unapply(diff(f,t$args[2]),t); fi; end: We also need the linalg and plots packages. with(linalg): with(plots):

Load the nonstardard packages above before you start this section. There are two Taylor expansions that we will work with here. The first and second order expansions. The first order expansion is given by 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, and the second order expansion is 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 . Using the above routines it is easy to calculate these expansions. Here is a simple example. The expansion is around the point (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). First we have a function and a point. f := x -> x[1]^2*x[2]^2 + 4*x[1]^2 + 4*x[2]^2; a := [1,1]; Now one can use the procedures fgrad and fhessain from above. Each routine takes two inputs, a function of a vector x and the dimension of x. gradf := fgrad(f,2); hessf := fhessian(f,2); Now we can get the two expansions. Taylor1 := unapply(f(a) + innerprod(gradf(a),[x[1],x[2]]-a),x); Taylor2 := unapply(Taylor1(x)+innerprod( [x[1],x[2]]-a,hessf(a)&*([x[1],x[2]]-a))/2,x); Here are plots of the function and the Taylor expansions. plotf := plot3d({f([x,y])},x=a[1]-2..a[1]+2, y=a[2]-2..a[2]+2,axes=boxed): plot_T1 := plot3d({Taylor1([x,y])},x=a[1]-2..a[1]+2, y=a[2]-2..a[2]+2, axes=boxed,color=blue): display(plotf,plot_T1); Here is the second order expansion plot. plot_T2 := plot3d({Taylor2([x,y])},x=a[1]-2..a[1]+2, y=a[2]-2..a[2]+2, axes=boxed,color=blue): display(plotf,plot_T2); Notice the better approximation with the second order Taylor polynomial.

Load the nonstandard packages above before you start. Consider the problem of minimizing a function 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 from NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHRkQvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZELyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1JJW1zdXBHRiU2JS1GLTY5USJSRihGMEYzRjZGOUY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZobkZbby1GLTY5USJuRihGMEYzRjZGOUY8Rj5GQEZCRkVGR0ZJRktGTUZPRlFGU0ZVRlhGWkZmbkZobkZbby8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRihGLDcjNiMpJSJSRyUibkc= to 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. If NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiMtSSNtaUdGJTY5USJmRigvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GKC8lJXNpemVHUSMxMkYoLyUlYm9sZEdRJmZhbHNlRigvJSdpdGFsaWNHUSV0cnVlRigvJSp1bmRlcmxpbmVHRjgvJSpzdWJzY3JpcHRHRjgvJSxzdXBlcnNjcmlwdEdGOC8lK2ZvcmVncm91bmRHUShbMCwwLDBdRigvJStiYWNrZ3JvdW5kR0ZELyUnb3BhcXVlR0Y4LyUrZXhlY3V0YWJsZUdGOC8lKXJlYWRvbmx5R0Y4LyUpY29tcG9zZWRHRjgvJSpjb252ZXJ0ZWRHRjgvJStpbXNlbGVjdGVkR0Y4LyUscGxhY2Vob2xkZXJHRjgvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RigvJSptYXRoY29sb3JHRkQvJS9tYXRoYmFja2dyb3VuZEdGRC8lK2ZvbnRmYW1pbHlHRjIvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YoLyUpbWF0aHNpemVHRjU3IzYjJSJmRw== has a local minimum at NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiMtSSNtaUdGJTY5USJhRigvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GKC8lJXNpemVHUSMxMkYoLyUlYm9sZEdRJmZhbHNlRigvJSdpdGFsaWNHUSV0cnVlRigvJSp1bmRlcmxpbmVHRjgvJSpzdWJzY3JpcHRHRjgvJSxzdXBlcnNjcmlwdEdGOC8lK2ZvcmVncm91bmRHUShbMCwwLDBdRigvJStiYWNrZ3JvdW5kR0ZELyUnb3BhcXVlR0Y4LyUrZXhlY3V0YWJsZUdGOC8lKXJlYWRvbmx5R0Y4LyUpY29tcG9zZWRHRjgvJSpjb252ZXJ0ZWRHRjgvJStpbXNlbGVjdGVkR0Y4LyUscGxhY2Vob2xkZXJHRjgvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RigvJSptYXRoY29sb3JHRkQvJS9tYXRoYmFja2dyb3VuZEdGRC8lK2ZvbnRmYW1pbHlHRjIvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YoLyUpbWF0aHNpemVHRjU3IzYjJSJhRw==, and one looks along any line 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, 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 has a minimum at 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. This means that 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 for all 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. One can restate this as the gradient of 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 at 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 is 0. This give the first order necessary condition for a minimum of a function. Consider the function 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. We want to find all points where the gradient is zero. One can use solve to find these points. This requires a little trickery. f := x -> x[1]^4-4*x[1]^2+x[2]^3-x[2]; gradf := fgrad(f,2); gradfx := gradf(x); sols1 := solve(convert(gradfx,set),{x[1],x[2]}); Note that the roots are not in usable form. One must use the allvalues command to find the roots. This takes a little work. crit_pts := map(allvalues,[sols1]); We are now ready to look at the second order condition. Here we look at the Hessian. Since the gradient is 0 at the critical points, if the Hessian is positive definite, the function looks like a parabolic bowl opening up. If the Hessian is negative definite, it opens down. If it is indeterminent, it is a saddle. Here is a check of the Hessian of all of the ctritical points. hess_f := fhessian(f,2); k := nops(crit_pts): for i from 1 to k do pt := subs(crit_pts[i],[x[1],x[2]]): print(` Critical point `,i,` is `,pt); h1_1 := evalf(hess_f(pt)[1,1]); det_h := evalf(det(hess_f(pt))); print(` The values for Sylvester's Theorem are `, h1_1,det_h); if h1_1>0 and det_h>0 then print(` It is a local min.`); print(`\n`); elif h1_1<0 and det_h>0 then print(` It is a local max.`); print(`\n`); elif det_h<0 then print(` It is a saddle.`); print(`\n`); else print(` There is no information.`); print(`\n`); fi: od: To check these results, we plot around critical point 6. a := subs(crit_pts[6],[x[1],x[2]]); plot3d(f([x,y]),x=a[1]-.2..a[1]+.2,y=a[2]-.2..a[2]+.2, axes=boxed);