Differential Properties of Curves

Two examples of closed curves:

Here are two vectors with components given in terms of the parameter t:

In[68]:=

PrettyFlower [ t_ ] := ( 1 4 + 3 4 Cos [ 3 t ] ) { Cos [ t ] ^ 3 , Sin [ t ] ^ 3 , Sin [ t ] Cos [ t ] ^ 2 }

Both of these functions define a list of 3 positions along each axis in terms of a single parameter

In[69]:=

Bendy [ t_ ] := { Cos [ t ] , Sin [ t ] , Sin [ t ] Cos [ t ] }

Defining Functions to Display Them

In[70]:=

showcurve [ VecFunc_ , tl_ ] := ParametricPlot3D [ Evaluate [ VecFunc [ tval ] ] , { tval , 0 , tl } , Compiled False , DisplayFunction Identity , PlotRange { { - 1 , 1 } , { - 1 , 1 } , { - 1 , 1 } } , BoxRatios { 1 , 1 , 1 } ]

In[71]:=

showline [ VecFunc_ , tl_ ] := Graphics3D [ { Thickness [ 0.01 ] , Hue [ 1 ] , Line [ { { 0 , 0 , 0 } , VecFunc [ tl ] } ] } ]

In[72]:=

showcurveline [ VecFunc_ , tl_ ] := Show [ { showcurve [ VecFunc , tl ] , showline [ VecFunc , tl ] } , DisplayFunction $DisplayFunction ]

In[73]:=

CurveLineSequence [ VecFunc_ ] := Table [ showcurveline [ VecFunc , i ] , { i , .1 , 3 Pi , .1 } ]

Animating the Curves with Their Parameter

The following command produces a sequence of 3D graphics objects that can be animated.  To animate: Select all the graphics objects that are output; use the "Cell" menu to access "Cell Grouping" and select "Open/Close Group" (this will collapse the output to a single plot and make your notebook less cluttered); then use the "Cell" menu again and select "Animate Selected Graphics."

In[74]:=

CurveLineSequence [ PrettyFlower ] ;

[Graphics:HTMLFiles/Lecture-12_95.gif]

Do the same thing for a different parameterized vector function:

In[75]:=

CurveLineSequence [ Bendy ] ;

[Graphics:HTMLFiles/Lecture-12_190.gif]

Calculating Arc Length and Re-parameterizing

Here is a general way to take a function of a general parameter, t, and compute the arc length traversed as t varies from one value to another:

In[76]:=

dFlowerDt = D [ Flower [ t ] , t ]

Out[76]=

Flower [ t ]

This is the arclength up to the parameter t

In[77]:=

sFlower = Integrate [ Sqrt [ dFlowerDt . dFlowerDt ] , t ]

General :: spell1 : Possible spelling error: new symbol name \" sFlower \" is similar to existing symbol \" Flower \". More… "Possible spelling error: new symbol name \\\"\\!\\(sFlower\\)\\\" is similar to existing symbol \\\"\\!\\(Flower\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, ButtonData:>\\\"General::spell1\\\"]\\)"

Out[77]=

Flower [ t ] . Flower [ t ] t

In other words , ds 2 = dx 2 + dy 2 + dz 2 so integrating the square root of this is the arclength

Applying this to the function Bendy defined above:

In[78]:=

dBendyDt = D [ Bendy [ t ] , t ]

Out[78]=

{ - Sin [ t ] , Cos [ t ] , Cos [ t ] 2 - Sin [ t ] 2 }

In[79]:=

sBendy = Integrate [ Sqrt [ dBendyDt . dBendyDt ] , t ]

General :: spell1 : Possible spelling error: new symbol name \" sBendy \" is similar to existing symbol \" Bendy \". More… "Possible spelling error: new symbol name \\\"\\!\\(sBendy\\)\\\" is similar to existing symbol \\\"\\!\\(Bendy\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, ButtonData:>\\\"General::spell1\\\"]\\)"

Out[79]=

EllipticE [ 2 t , 1 2 ] 2

The arc length in this case is given by a tabulated  function called an elliptic integral and after checking its behavior at t = 0 we can plot it over the range {t,0,2π}:

In[80]:=

sBendy /. t 0

Out[80]=

0

In[81]:=

Plot [ sBendy , { t , 0 , 2 Pi } ]

[Graphics:HTMLFiles/Lecture-12_191.gif]

Out[81]=

Graphics

Alternatively, we can evaluate the expression for arc length numberically using the following:

In[82]:=

Plot [ Evaluate [ NIntegrate [ Sqrt [ dBendyDt . dBendyDt ] , { t , 0 , uplim } ] ] , { uplim , 0 , 6.4 } ]

NIntegrate :: nlim : t = uplim is not a valid limit of integration. More… "\\!\\(t\\) = \\!\\(uplim\\) is not a valid limit of integration. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, ButtonData:>\\\"NIntegrate::nlim\\\"]\\)"

[Graphics:HTMLFiles/Lecture-12_192.gif]

Out[82]=

Graphics

In[83]:=

FlowerPot [ u_ , v_ ] := { ( 3 + Cos [ v ] ) Cos [ u ] , Sin [ u ] + ( 3 + Cos [ v ] ) Sin [ u ] , ( 3 / 2 + Cos [ u + v ] ) Sin [ v ] }

Example of a parameritized surface

In[84]:=

<< Graphics`ParametricPlot3D`

In[85]:=

Flowerplot = ParametricPlot3D [ FlowerPot [ u , v ] , { u , 0 , 2 Pi } , { v , 0 , 2 Pi } , ViewPoint -> { 0.141 , 1.653 , 1.117 } , PlotPoints -> { 120 , 40 } ]

[Graphics:HTMLFiles/Lecture-12_193.gif]

Out[85]=

Graphics3D

A Curve on a parameritized surface

In[86]:=

Vines[t_] := 1.025 * FlowerPot[t Cos[t], -t^2Sin[ t]]

vineplot = ParametricPlot3D[Vines[t], {t, 0, 2 Pi}, ViewPoint-> {0.141, 1.653, 1.117}, PlotPoints->500]

[Graphics:HTMLFiles/Lecture-12_196.gif]

Out[87]=

Graphics3D

In the above, I want to put the vine outside the surface so I scale it out a little bit...

Out[53]=

Graphics3D

In[88]:=

thickvineplot = Show [ { Graphics3D [ Thickness [ 0.02 ] ] , Graphics3D [ Hue [ 0.333 , 0.5 , 0.5 ] ] , vineplot } ]

[Graphics:HTMLFiles/Lecture-12_197.gif]

Out[88]=

Graphics3D

In[89]:=

Show [ thickvineplot , Flowerplot ]

[Graphics:HTMLFiles/Lecture-12_198.gif]

Out[89]=

Graphics3D

In[90]:=

Show [ Flowerplot , thickvineplot ]

[Graphics:HTMLFiles/Lecture-12_199.gif]

Out[90]=

Graphics3D

Multivariable Calculus: Mathematica Review

AScalarFunction is defined everywhere in (x,y,z)

In[91]:=

AScalarFunction [ x_ , y_ , z_ ] := SomeFunction [ x , y , z ]

In[92]:=

AScalarFunction [ x , y , z ]

Out[92]=

SomeFunction [ x , y , z ]

The following lines print and they define expressions.

In[93]:=

Print["derivative w/r to first argument is " ] ;   dFuncX = D[AScalarFunction[x, y, z], x]

Print["derivative w/r to second argument is " ] ;   dFuncY = D[AScalarFunction[x, y, z], y]

Print["derivative w/r to third argument is " ] ;   dFuncZ = D[AScalarFunction[x, y, z], z]

derivative w/r to first argument is

Out[93]=

SomeFunction ( 1 , 0 , 0 ) [ x , y , z ]

General :: spell1 : Possible spelling error: new symbol name \" dFuncY \" is similar to existing symbol \" dFuncX \". More… "Possible spelling error: new symbol name \\\"\\!\\(dFuncY\\)\\\" is similar to existing symbol \\\"\\!\\(dFuncX\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, ButtonData:>\\\"General::spell1\\\"]\\)"

derivative w/r to second argument is

Out[94]=

SomeFunction ( 0 , 1 , 0 ) [ x , y , z ]

General :: spell : Possible spelling error: new symbol name \" dFuncZ \" is similar to existing symbols { dFuncX , dFuncY } . More… "Possible spelling error: new symbol name \\\"\\!\\(dFuncZ\\)\\\" is similar to existing symbols \\!\\({dFuncX, dFuncY}\\). \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, ButtonData:>\\\"General::spell\\\"]\\)"

derivative w/r to third argument is

Out[95]=

SomeFunction ( 0 , 0 , 1 ) [ x , y , z ]

In the output lines above, partial derivatives are denoted by superscripts: e.g., (1,0,0) indicates the first partial derivative with respect to the first variable, x. The second partial with respect to y and z would be denoted by a superscript (0,1,1).

x(w,v), y(w,v), z(w,v) is a restriction of all space to a surface parameterized by (w,v),
AScalarFunction is now defined on the surface as a function of (w,v)

In[96]:=

AScalarFunction [ x [ w , v ] , y [ w , v ] , z [ w , v ] ]

Out[96]=

SomeFunction [ x [ w , v ] , y [ w , v ] , z [ w , v ] ]

Because it is now a function of w and v, the derivative with respect to x will vanish:

In[97]:=

D [ AScalarFunction [ x [ w , v ] , y [ w , v ] , z [ w , v ] ] , x ]

Out[97]=

0

Two more flavors of derivatives, these are partial derivatives evaluated on the surface

In[98]:=

dFuncW = D [ AScalarFunction [ x [ w , v ] , y [ w , v ] , z [ w , v ] ] , w ]

General :: spell : Possible spelling error: new symbol name \" dFuncW \" is similar to existing symbols { dFuncX , dFuncY , dFuncZ } . More… "Possible spelling error: new symbol name \\\"\\!\\(dFuncW\\)\\\" is similar to existing symbols \\!\\({dFuncX, dFuncY, dFuncZ}\\). \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, ButtonData:>\\\"General::spell\\\"]\\)"

Out[98]=

z ( 1 , 0 ) [ w , v ] SomeFunction ( 0 , 0 , 1 ) [ x [ w , v ] , y [ w , v ] , z [ w , v ] ] + y ( 1 , 0 ) [ w , v ] SomeFunction ( 0 , 1 , 0 ) [ x [ w , v ] , y [ w , v ] , z [ w , v ] ] + x ( 1 , 0 ) [ w , v ] SomeFunction ( 1 , 0 , 0 ) [ x [ w , v ] , y [ w , v ] , z [ w , v ] ]

In[99]:=

dFuncV = D [ AScalarFunction [ x [ w , v ] , y [ w , v ] , z [ w , v ] ] , v ]

General :: spell : Possible spelling error: new symbol name \" dFuncV \" is similar to existing symbols { dFuncW , dFuncX , dFuncY , dFuncZ } . More… "Possible spelling error: new symbol name \\\"\\!\\(dFuncV\\)\\\" is similar to existing symbols \\!\\({dFuncW, dFuncX, dFuncY, dFuncZ}\\). \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, ButtonData:>\\\"General::spell\\\"]\\)"

Out[99]=

z ( 0 , 1 ) [ w , v ] SomeFunction ( 0 , 0 , 1 ) [ x [ w , v ] , y [ w , v ] , z [ w , v ] ] + y ( 0 , 1 ) [ w , v ] SomeFunction ( 0 , 1 , 0 ) [ x [ w , v ] , y [ w , v ] , z [ w , v ] ] + x ( 0 , 1 ) [ w , v ] SomeFunction ( 1 , 0 , 0 ) [ x [ w , v ] , y [ w , v ] , z [ w , v ] ]

On the surface x(w,v), y(w,v), z(w,v), we can prescribe a curve w(t), v(t), now we have
AScalarFunction defined on that curve

In[100]:=

AScalarFunction [ x [ w [ t ] , v [ t ] ] , y [ w [ t ] , v [ t ] ] , z [ w [ t ] , v [ t ] ] ]

Out[100]=

SomeFunction [ x [ w [ t ] , v [ t ] ] , y [ w [ t ] , v [ t ] ] , z [ w [ t ] , v [ t ] ] ]

The following is a derivative of the function along the curve parameterized by t

In[101]:=

dFuncT = D [ AScalarFunction [ x [ w [ t ] , v [ t ] ] , y [ w [ t ] , v [ t ] ] , z [ w [ t ] , v [ t ] ] ] , t ]

General :: spell : Possible spelling error: new symbol name \" dFuncT \" is similar to existing symbols { dFuncV , dFuncW , dFuncX , dFuncY , dFuncZ } . More… "Possible spelling error: new symbol name \\\"\\!\\(dFuncT\\)\\\" is similar to existing symbols \\!\\({dFuncV, dFuncW, dFuncX, dFuncY, dFuncZ}\\). \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, ButtonData:>\\\"General::spell\\\"]\\)"

Out[101]=

( v [ t ] z ( 0 , 1 ) [ w [ t ] , v [ t ] ] + w [ t ] z ( 1 , 0 ) [ w [ t ] , v [ t ] ] ) SomeFunction ( 0 , 0 , 1 ) [ x [ w [ t ] , v [ t ] ] , y [ w [ t ] , v [ t ] ] , z [ w [ t ] , v [ t ] ] ] + ( v [ t ] y ( 0 , 1 ) [ w [ t ] , v [ t ] ] + w [ t ] y ( 1 , 0 ) [ w [ t ] , v [ t ] ] ) SomeFunction ( 0 , 1 , 0 ) [ x [ w [ t ] , v [ t ] ] , y [ w [ t ] , v [ t ] ] , z [ w [ t ] , v [ t ] ] ] + ( v [ t ] x ( 0 , 1 ) [ w [ t ] , v [ t ] ] + w [ t ] x ( 1 , 0 ) [ w [ t ] , v [ t ] ] ) SomeFunction ( 1 , 0 , 0 ) [ x [ w [ t ] , v [ t ] ] , y [ w [ t ] , v [ t ] ] , z [ w [ t ] , v [ t ] ] ]

Note on the step immediately above: by specifying w and v, values of x and y are specified, and additionally values of z.  The three functions x(w,v), y(w,v), and z(w,v) together describe a surface---it specifies that three points can be specified by two values (A familiar case is when w=x, and v=y, then z(x,y) is a surface that can be specified over the x-y plane. The functions w(t) and v(t) trace out a plane curve on the w-v surface and map onto a corresponding twisted curve (see Kreyszig p.429 for distinction between "plane" and "twisted" if it is not obvious).

Finally, we could skip the surface and just define a space curve x(t), y(t), z(t) and  take the derivative of AScalarFunc along  that curve:

In[102]:=

dFuncT = D [ AScalarFunction [ x [ t ] , y [ t ] , z [ t ] ] , t ]

Out[102]=

z [ t ] SomeFunction ( 0 , 0 , 1 ) [ x [ t ] , y [ t ] , z [ t ] ] + y [ t ] SomeFunction ( 0 , 1 , 0 ) [ x [ t ] , y [ t ] , z [ t ] ] + x [ t ] SomeFunction ( 1 , 0 , 0 ) [ x [ t ] , y [ t ] , z [ t ] ]

Here, dF is interpreted as any change in the scalar function if its variables are also changed dx, dy, dz

Visualizing approximations (Taylor Expansions)  to surfaces at points

Getting Mathematica to represent a general change in a function--multidimensional versions of Taylor expansions

In[103]:=

SmallChangeSeries = Expand [ Series [ AScalarFunction [ x + dx , y + dy , z + dz ] , { dx , 0 , 1 } , { dy , 0 , 1 } , { dz , 0 , 1 } ] ] - AScalarFunction [ x , y , z ]

Out[103]=

( ( SomeFunction ( 0 , 0 , 1 ) [ x , y , z ] dz + O [ dz ] 2 SeriesData dz 0 1 2 1 SeriesData dz 0 0 0 1 SomeFunction x y z 1 2 1 ) + ( SomeFunction ( 0 , 1 , 0 ) [ x , y , z ] + SomeFunction ( 0 , 1 , 1 ) [ x , y , z ] dz + O [ dz ] 2 SeriesData dz 0 0 2 1 SeriesData dz 0 0 1 0 SomeFunction x y z 0 1 1 SomeFunction x y z 0 2 1 ) dy + O [ dy ] 2 SeriesData dy 0 0 2 1 SeriesData dy 0 SeriesData dz 0 0 0 1 SomeFunction x y z 1 2 1 SeriesData dz 0 0 1 0 SomeFunction x y z 0 1 1 SomeFunction x y z 0 2 1 0 2 1 ) + ( ( SomeFunction ( 1 , 0 , 0 ) [ x , y , z ] + SomeFunction ( 1 , 0 , 1 ) [ x , y , z ] dz + O [ dz ] 2 SeriesData dz 0 0 2 1 SeriesData dz 0 1 0 0 SomeFunction x y z 1 0 1 SomeFunction x y z 0 2 1 ) + ( SomeFunction ( 1 , 1 , 0 ) [ x , y , z ] + SomeFunction ( 1 , 1 , 1 ) [ x , y , z ] dz + O [ dz ] 2 SeriesData dz 0 0 2 1 SeriesData dz 0 1 1 0 SomeFunction x y z 1 1 1 SomeFunction x y z 0 2 1 ) dy + O [ dy ] 2 SeriesData dy 0 0 2 1 SeriesData dy 0 SeriesData dz 0 1 0 0 SomeFunction x y z 1 0 1 SomeFunction x y z 0 2 1 SeriesData dz 0 1 1 0 SomeFunction x y z 1 1 1 SomeFunction x y z 0 2 1 0 2 1 ) dx + O [ dx ] 2 SeriesData dx 0 0 2 1 SeriesData dx 0 SeriesData dy 0 SeriesData dz 0 0 0 1 SomeFunction x y z 1 2 1 SeriesData dz 0 0 1 0 SomeFunction x y z 0 1 1 SomeFunction x y z 0 2 1 0 2 1 SeriesData dy 0 SeriesData dz 0 1 0 0 SomeFunction x y z 1 0 1 SomeFunction x y z 0 2 1 SeriesData dz 0 1 1 0 SomeFunction x y z 1 1 1 SomeFunction x y z 0 2 1 0 2 1 0 2 1

In[104]:=

dScalarFunction = Expand [ Normal [ SmallChangeSeries ] ]

Out[104]=

dz SomeFunction ( 0 , 0 , 1 ) [ x , y , z ] + dy SomeFunction ( 0 , 1 , 0 ) [ x , y , z ] + dy dz SomeFunction ( 0 , 1 , 1 ) [ x , y , z ] + dx SomeFunction ( 1 , 0 , 0 ) [ x , y , z ] + dx dz SomeFunction ( 1 , 0 , 1 ) [ x , y , z ] + dx dy SomeFunction ( 1 , 1 , 0 ) [ x , y , z ] + dx dy dz SomeFunction ( 1 , 1 , 1 ) [ x , y , z ]

The next step eliminates second- and third-order terms… (remember, dx, dy, and dz are small)

In[105]:=

dScalarFunction = dScalarFunction /. { dx dy 0 , dy dz 0 , dx dz 0 }

Out[105]=

dz SomeFunction ( 0 , 0 , 1 ) [ x , y , z ] + dy SomeFunction ( 0 , 1 , 0 ) [ x , y , z ] + dx SomeFunction ( 1 , 0 , 0 ) [ x , y , z ]

The above form is like the thermodynamic expression : dF = F x dx + F y dy + F z dz

Example of using Taylor Expansion to turn a function of two variables into an approximating function of four  variables:

In[106]:=

CrazyFun [ x_ , y_ ] := Sin [ 5 π x ] Sin [ 5 π y ] x y + Sin [ 5 π ( x - 1 ) ] Sin [ 5 π ( y - 1 ) ] ( x - 1 ) ( y - 1 )

Plot this function over suitable range of the variables x and y:

In[107]:=

theplot = Plot3D [ CrazyFun [ x , y ] , { x , 0.1 , .9 } , { y , 0.1 , .9 } , PlotRange All , Mesh False ]

[Graphics:HTMLFiles/Lecture-12_205.gif]

Out[107]=

SurfaceGraphics

Now, approximate the function about a specific point (xo, yo), using Mathematica's Series function:

In[108]:=

Approxfunction [ x_ , y_ , xo_ , yo_ ] := Series [ CrazyFun [ x , y ] , { x , xo , 2 } , { y , yo , 2 } ] // Normal

and plot the approximate function in the neighborhood of (xo, yo):

In[109]:=

anapprox = Plot3D [ Evaluate [ Approxfunction [ x , y , .7 , .1 ] ] , { x , .7 - .1 , .7 + .1 } , { y , .1 - .1 , .1 + .1 } ]

[Graphics:HTMLFiles/Lecture-12_206.gif]

Out[109]=

SurfaceGraphics

Both the original and the approximate function can be plotted simultaneously:

In[110]:=

Show [ anapprox , theplot ]

[Graphics:HTMLFiles/Lecture-12_207.gif]

Out[110]=

Graphics3D

This final little bit selects random points (xo, yo) and fits Taylor expansions to ten different points, then displays them individually as well as with a superposed plot of the original function.

In[111]:=

Table [ { xo [ i ] = Random [ ] , yo [ i ] = Random [ ] } , { i , 1 , 100 } ] ;

The next function automates the small approximating surface patch

In[112]:=

ApproxPlot [ i_ ] := Plot3D [ Evaluate [ Approxfunction [ x , y , xo [ i ] , yo [ i ] ] ] , { x , xo [ i ] - .1 , xo [ i ] + .1 } , { y , yo [ i ] - .1 , yo [ i ] + .1 } , PlotPoints 6 , ColorFunction ( RGBColor [ xo [ i ] , yo [ i ] , # ] & ) , DisplayFunction Identity ]

To build a sequence of graphics, I'll build a stack of ten graphics objects by using a recursive method. The next command sets the end of the recursion loop.

In[113]:=

Clear [ GraphicsStack ]

In[114]:=

GraphicsStack [ 0 ] = Show [ ApproxPlot [ 1 ] , DisplayFunction Identity ]

Out[114]=

SurfaceGraphics

Here is the recursive function.

In[115]:=

GraphicsStack [ i_ ] := GraphicsStack [ i ] = Show [ GraphicsStack [ i - 1 ] , ApproxPlot [ i + 1 ] ]

In[116]:=

Show [ GraphicsArray [ { GraphicsStack [ 10 ] , Show [ theplot , GraphicsStack [ 10 ] ] } ] , DisplayFunction $DisplayFunction ]

Graphics :: realu : Argument in RGBColor [ 0.1947701615072844 , 0.1938584747649562 , 1.0083416833631114 ] is not a real number between 0 and 1. More… "Argument in \\!\\(RGBColor[\\(\\(0.1947701615072844`, 0.1938584747649562`, 1.0083416833631114`\\)\\)]\\) is not a real number between 0 and 1. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, ButtonData:>\\\"Graphics::realu\\\"]\\)"

Graphics :: realu : Argument in RGBColor [ 0.1947701615072844 , 0.1938584747649562 , 1.0083416833631114 ] is not a real number between 0 and 1. More… "Argument in \\!\\(RGBColor[\\(\\(0.1947701615072844`, 0.1938584747649562`, 1.0083416833631114`\\)\\)]\\) is not a real number between 0 and 1. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, ButtonData:>\\\"Graphics::realu\\\"]\\)"

Graphics :: realu : Argument in RGBColor [ 0.1947701615072844 , 0.1938584747649562 , 1.0083416833631114 ] is not a real number between 0 and 1. More… "Argument in \\!\\(RGBColor[\\(\\(0.1947701615072844`, 0.1938584747649562`, 1.0083416833631114`\\)\\)]\\) is not a real number between 0 and 1. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, ButtonData:>\\\"Graphics::realu\\\"]\\)"

General :: stop : Further output of Graphics :: realu will be suppressed during this calculation. More… "Further output of \\!\\(Graphics :: \\\"realu\\\"\\) will be suppressed during this calculation. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, ButtonData:>\\\"General::stop\\\"]\\)"

[Graphics:HTMLFiles/Lecture-12_208.gif]

[Graphics:HTMLFiles/Lecture-12_209.gif]

Out[116]=

GraphicsArray


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