Differential Properties of Curves

Two examples of closed curves:

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

In[68]:=

$\mathrm{PrettyFlower}\left[\text{t\_}\right]:=\left(\frac{1}{4}+\frac{3}{4}\mathrm{Cos}\left[3t\right]\right)\left\{\mathrm{Cos}\left[t\right]^3,\mathrm{Sin}\left[t\right]^3,\mathrm{Sin}\left[t\right]\mathrm{Cos}\left[t\right]^2\right\}$

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

In[69]:=

$\mathrm{Bendy}\left[\text{t\_}\right]:=\left\{\mathrm{Cos}\left[t\right],\mathrm{Sin}\left[t\right],\mathrm{Sin}\left[t\right]\mathrm{Cos}\left[t\right]\right\}$

Defining Functions to Display Them

In[70]:=

$\mathrm{showcurve}\left[\text{VecFunc\_},\text{tl\_}\right]:=\mathrm{ParametricPlot3D}\left[\mathrm{Evaluate}\left[\mathrm{VecFunc}\left[\mathrm{tval}\right]\right],\left\{\mathrm{tval},0,\mathrm{tl}\right\},\mathrm{Compiled}\to \mathrm{False},\text{}\mathrm{DisplayFunction}\to \mathrm{Identity},\mathrm{PlotRange}\to \left\{\left\{-1,1\right\},\left\{-1,1\right\},\left\{-1,1\right\}\right\},\mathrm{BoxRatios}\to \left\{1,1,1\right\}\right]$

In[71]:=

$\mathrm{showline}\left[\text{VecFunc\_},\text{tl\_}\right]:=\mathrm{Graphics3D}\left[\left\{\mathrm{Thickness}\left[0.01\right],\mathrm{Hue}\left[1\right],\mathrm{Line}\left[\left\{\left\{0,0,0\right\},\mathrm{VecFunc}\left[\mathrm{tl}\right]\right\}\right]\right\}\right]$

In[72]:=

$\mathrm{showcurveline}\left[\text{VecFunc\_},\text{tl\_}\right]:=\mathrm{Show}\left[\left\{\mathrm{showcurve}\left[\mathrm{VecFunc},\mathrm{tl}\right],\mathrm{showline}\left[\mathrm{VecFunc},\mathrm{tl}\right]\right\},\mathrm{DisplayFunction}\to \mathrm{\$DisplayFunction}\right]$

In[73]:=

$\mathrm{CurveLineSequence}\left[\text{VecFunc\_}\right]:=\mathrm{Table}\left[\mathrm{showcurveline}\left[\mathrm{VecFunc},i\right],\left\{i,\text{.1},3\mathrm{Pi},\text{.1}\right\}\right]$

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]:=

$\mathrm{CurveLineSequence}\left[\mathrm{PrettyFlower}\right];$

Do the same thing for a different parameterized vector function:

In[75]:=

$\mathrm{CurveLineSequence}\left[\mathrm{Bendy}\right];$

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]:=

$\mathrm{dFlowerDt}=D\left[\mathrm{Flower}\left[t\right],t\right]$

Out[76]=

${\mathrm{Flower}}^{\prime }\left[t\right]$

This is the arclength up to the parameter t

In[77]:=

$\mathrm{sFlower}=\mathrm{Integrate}\left[\mathrm{Sqrt}\left[\mathrm{dFlowerDt}.\mathrm{dFlowerDt}\right],t\right]$

$\mathrm{General}\text{::}\mathrm{spell1}:\text{Possible spelling error: new symbol name "}\mathrm{sFlower}\text{" is similar to existing symbol "}\mathrm{Flower}\text{".}\mathrm{More\dots }$

Out[77]=

$\int \sqrt{{\mathrm{Flower}}^{\prime }\left[t\right].{\mathrm{Flower}}^{\prime }\left[t\right]}dt$

$\mathrm{In}\mathrm{other}\mathrm{words},{\mathrm{ds}}^2={\mathrm{dx}}^2+{\mathrm{dy}}^2+{\mathrm{dz}}^2\mathrm{so}\mathrm{integrating}\mathrm{the}\mathrm{square}\mathrm{root}\mathrm{of}\mathrm{this}\mathrm{is}\mathrm{the}\mathrm{arclength}$

Applying this to the function Bendy defined above:

In[78]:=

$\mathrm{dBendyDt}=D\left[\mathrm{Bendy}\left[t\right],t\right]$

Out[78]=

$\left\{-\mathrm{Sin}\left[t\right],\mathrm{Cos}\left[t\right],{\mathrm{Cos}\left[t\right]}^2-{\mathrm{Sin}\left[t\right]}^2\right\}$

In[79]:=

$\mathrm{sBendy}=\mathrm{Integrate}\left[\mathrm{Sqrt}\left[\mathrm{dBendyDt}.\mathrm{dBendyDt}\right],t\right]$

$\mathrm{General}\text{::}\mathrm{spell1}:\text{Possible spelling error: new symbol name "}\mathrm{sBendy}\text{" is similar to existing symbol "}\mathrm{Bendy}\text{".}\mathrm{More\dots }$

Out[79]=

$\frac{\mathrm{EllipticE}\left[2t,\frac{1}{2}\right]}{\sqrt{2}}$

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]:=

$\mathrm{sBendy}/.t\to 0$

Out[80]=

$0$

In[81]:=

$\mathrm{Plot}\left[\mathrm{sBendy},\left\{t,0,2\mathrm{Pi}\right\}\right]$

Out[81]=

$⁃\mathrm{Graphics}⁃$

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

In[82]:=

$\mathrm{Plot}\left[\mathrm{Evaluate}\left[\mathrm{NIntegrate}\left[\mathrm{Sqrt}\left[\mathrm{dBendyDt}.\mathrm{dBendyDt}\right],\left\{t,0,\mathrm{uplim}\right\}\right]\right],\left\{\mathrm{uplim},0,6.4\right\}\right]$

$\mathrm{NIntegrate}\text{::}\mathrm{nlim}:t\text{=}\mathrm{uplim}\text{is not a valid limit of integration.}\mathrm{More\dots }$

Out[82]=

$⁃\mathrm{Graphics}⁃$

In[83]:=

$\mathrm{FlowerPot}\left[\text{u\_},\text{v\_}\right]:=\left\{\left(3+\mathrm{Cos}\left[v\right]\right)\mathrm{Cos}\left[u\right],\mathrm{Sin}\left[u\right]+\left(3+\mathrm{Cos}\left[v\right]\right)\mathrm{Sin}\left[u\right],\left(3/2+\mathrm{Cos}\left[u+v\right]\right)\mathrm{Sin}\left[v\right]\right\}$

Example of a parameritized surface

In[84]:=

$<<\text{Graphics`ParametricPlot3D`}$

In[85]:=

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

Out[85]=

$⁃\mathrm{Graphics3D}⁃$

A Curve on a parameritized surface

In[86]:=

Out[87]=

$⁃\mathrm{Graphics3D}⁃$

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

Out[53]=

$⁃\mathrm{Graphics3D}⁃$

In[88]:=

$\mathrm{thickvineplot}=\mathrm{Show}\left[\left\{\mathrm{Graphics3D}\left[\mathrm{Thickness}\left[0.02\right]\right],\mathrm{Graphics3D}\left[\mathrm{Hue}\left[0.333,0.5,0.5\right]\right],\mathrm{vineplot}\right\}\right]$

Out[88]=

$⁃\mathrm{Graphics3D}⁃$

In[89]:=

$\mathrm{Show}\left[\mathrm{thickvineplot},\mathrm{Flowerplot}\right]$

Out[89]=

$⁃\mathrm{Graphics3D}⁃$

In[90]:=

$\mathrm{Show}\left[\mathrm{Flowerplot},\mathrm{thickvineplot}\right]$

Out[90]=

$⁃\mathrm{Graphics3D}⁃$

Multivariable Calculus: Mathematica Review

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

In[91]:=

$\mathrm{AScalarFunction}\left[\text{x\_},\text{y\_},\text{z\_}\right]:=\mathrm{SomeFunction}\left[x,y,z\right]$

In[92]:=

$\mathrm{AScalarFunction}\left[x,y,z\right]$

Out[92]=

$\mathrm{SomeFunction}\left[x,y,z\right]$

The following lines print and they define expressions.

In[93]:=

$\text{derivative w/r to first argument is}$

Out[93]=

${\mathrm{SomeFunction}}^{\left(1,0,0\right)}\left[x,y,z\right]$

$\mathrm{General}\text{::}\mathrm{spell1}:\text{Possible spelling error: new symbol name "}\mathrm{dFuncY}\text{" is similar to existing symbol "}\mathrm{dFuncX}\text{".}\mathrm{More\dots }$

$\text{derivative w/r to second argument is}$

Out[94]=

${\mathrm{SomeFunction}}^{\left(0,1,0\right)}\left[x,y,z\right]$

$\mathrm{General}\text{::}\mathrm{spell}:\text{Possible spelling error: new symbol name "}\mathrm{dFuncZ}\text{" is similar to existing symbols}\left\{\mathrm{dFuncX},\mathrm{dFuncY}\right\}\text{.}\mathrm{More\dots }$

$\text{derivative w/r to third argument is}$

Out[95]=

${\mathrm{SomeFunction}}^{\left(0,0,1\right)}\left[x,y,z\right]$

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]:=

$\mathrm{AScalarFunction}\left[x\left[w,v\right],y\left[w,v\right],z\left[w,v\right]\right]$

Out[96]=

$\mathrm{SomeFunction}\left[x\left[w,v\right],y\left[w,v\right],z\left[w,v\right]\right]$

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

In[97]:=

$D\left[\mathrm{AScalarFunction}\left[x\left[w,v\right],y\left[w,v\right],z\left[w,v\right]\right],x\right]$

Out[97]=

$0$

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

In[98]:=

$\mathrm{dFuncW}=D\left[\mathrm{AScalarFunction}\left[x\left[w,v\right],y\left[w,v\right],z\left[w,v\right]\right],w\right]$

$\mathrm{General}\text{::}\mathrm{spell}:\text{Possible spelling error: new symbol name "}\mathrm{dFuncW}\text{" is similar to existing symbols}\left\{\mathrm{dFuncX},\mathrm{dFuncY},\mathrm{dFuncZ}\right\}\text{.}\mathrm{More\dots }$

Out[98]=

$z^{\left(1,0\right)}\left[w,v\right]{\mathrm{SomeFunction}}^{\left(0,0,1\right)}\left[x\left[w,v\right],y\left[w,v\right],z\left[w,v\right]\right]+y^{\left(1,0\right)}\left[w,v\right]{\mathrm{SomeFunction}}^{\left(0,1,0\right)}\left[x\left[w,v\right],y\left[w,v\right],z\left[w,v\right]\right]+x^{\left(1,0\right)}\left[w,v\right]{\mathrm{SomeFunction}}^{\left(1,0,0\right)}\left[x\left[w,v\right],y\left[w,v\right],z\left[w,v\right]\right]$

In[99]:=

$\mathrm{dFuncV}=D\left[\mathrm{AScalarFunction}\left[x\left[w,v\right],y\left[w,v\right],z\left[w,v\right]\right],v\right]$

$\mathrm{General}\text{::}\mathrm{spell}:\text{Possible spelling error: new symbol name "}\mathrm{dFuncV}\text{" is similar to existing symbols}\left\{\mathrm{dFuncW},\mathrm{dFuncX},\mathrm{dFuncY},\mathrm{dFuncZ}\right\}\text{.}\mathrm{More\dots }$

Out[99]=

$z^{\left(0,1\right)}\left[w,v\right]{\mathrm{SomeFunction}}^{\left(0,0,1\right)}\left[x\left[w,v\right],y\left[w,v\right],z\left[w,v\right]\right]+y^{\left(0,1\right)}\left[w,v\right]{\mathrm{SomeFunction}}^{\left(0,1,0\right)}\left[x\left[w,v\right],y\left[w,v\right],z\left[w,v\right]\right]+x^{\left(0,1\right)}\left[w,v\right]{\mathrm{SomeFunction}}^{\left(1,0,0\right)}\left[x\left[w,v\right],y\left[w,v\right],z\left[w,v\right]\right]$

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]:=

$\mathrm{AScalarFunction}\left[x\left[w\left[t\right],v\left[t\right]\right],y\left[w\left[t\right],v\left[t\right]\right],z\left[w\left[t\right],v\left[t\right]\right]\right]$

Out[100]=

$\mathrm{SomeFunction}\left[x\left[w\left[t\right],v\left[t\right]\right],y\left[w\left[t\right],v\left[t\right]\right],z\left[w\left[t\right],v\left[t\right]\right]\right]$

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

In[101]:=

$\mathrm{dFuncT}=D\left[\mathrm{AScalarFunction}\left[x\left[w\left[t\right],v\left[t\right]\right],y\left[w\left[t\right],v\left[t\right]\right],z\left[w\left[t\right],v\left[t\right]\right]\right],t\right]$

$\mathrm{General}\text{::}\mathrm{spell}:\text{Possible spelling error: new symbol name "}\mathrm{dFuncT}\text{" is similar to existing symbols}\left\{\mathrm{dFuncV},\mathrm{dFuncW},\mathrm{dFuncX},\mathrm{dFuncY},\mathrm{dFuncZ}\right\}\text{.}\mathrm{More\dots }$

Out[101]=

$\left(v^{\prime }\left[t\right]z^{\left(0,1\right)}\left[w\left[t\right],v\left[t\right]\right]+w^{\prime }\left[t\right]z^{\left(1,0\right)}\left[w\left[t\right],v\left[t\right]\right]\right){\mathrm{SomeFunction}}^{\left(0,0,1\right)}\left[x\left[w\left[t\right],v\left[t\right]\right],y\left[w\left[t\right],v\left[t\right]\right],z\left[w\left[t\right],v\left[t\right]\right]\right]+\left(v^{\prime }\left[t\right]y^{\left(0,1\right)}\left[w\left[t\right],v\left[t\right]\right]+w^{\prime }\left[t\right]y^{\left(1,0\right)}\left[w\left[t\right],v\left[t\right]\right]\right){\mathrm{SomeFunction}}^{\left(0,1,0\right)}\left[x\left[w\left[t\right],v\left[t\right]\right],y\left[w\left[t\right],v\left[t\right]\right],z\left[w\left[t\right],v\left[t\right]\right]\right]+\left(v^{\prime }\left[t\right]x^{\left(0,1\right)}\left[w\left[t\right],v\left[t\right]\right]+w^{\prime }\left[t\right]x^{\left(1,0\right)}\left[w\left[t\right],v\left[t\right]\right]\right){\mathrm{SomeFunction}}^{\left(1,0,0\right)}\left[x\left[w\left[t\right],v\left[t\right]\right],y\left[w\left[t\right],v\left[t\right]\right],z\left[w\left[t\right],v\left[t\right]\right]\right]$

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]:=

$\mathrm{dFuncT}=D\left[\mathrm{AScalarFunction}\left[x\left[t\right],y\left[t\right],z\left[t\right]\right],t\right]$

Out[102]=

$z^{\prime }\left[t\right]{\mathrm{SomeFunction}}^{\left(0,0,1\right)}\left[x\left[t\right],y\left[t\right],z\left[t\right]\right]+y^{\prime }\left[t\right]{\mathrm{SomeFunction}}^{\left(0,1,0\right)}\left[x\left[t\right],y\left[t\right],z\left[t\right]\right]+x^{\prime }\left[t\right]{\mathrm{SomeFunction}}^{\left(1,0,0\right)}\left[x\left[t\right],y\left[t\right],z\left[t\right]\right]$

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]:=

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

Out[103]=

$\left(\left({\mathrm{SomeFunction}}^{\left(0,0,1\right)}\left[x,y,z\right]\mathrm{dz}+{O\left[\mathrm{dz}\right]}^2\right)+\left({\mathrm{SomeFunction}}^{\left(0,1,0\right)}\left[x,y,z\right]+{\mathrm{SomeFunction}}^{\left(0,1,1\right)}\left[x,y,z\right]\mathrm{dz}+{O\left[\mathrm{dz}\right]}^2\right)\mathrm{dy}+{O\left[\mathrm{dy}\right]}^2\right)+\left(\left({\mathrm{SomeFunction}}^{\left(1,0,0\right)}\left[x,y,z\right]+{\mathrm{SomeFunction}}^{\left(1,0,1\right)}\left[x,y,z\right]\mathrm{dz}+{O\left[\mathrm{dz}\right]}^2\right)+\left({\mathrm{SomeFunction}}^{\left(1,1,0\right)}\left[x,y,z\right]+{\mathrm{SomeFunction}}^{\left(1,1,1\right)}\left[x,y,z\right]\mathrm{dz}+{O\left[\mathrm{dz}\right]}^2\right)\mathrm{dy}+{O\left[\mathrm{dy}\right]}^2\right)\mathrm{dx}+{O\left[\mathrm{dx}\right]}^2$

In[104]:=

$\mathrm{dScalarFunction}=\mathrm{Expand}\left[\mathrm{Normal}\left[\mathrm{SmallChangeSeries}\right]\right]$

Out[104]=

$\mathrm{dz}{\mathrm{SomeFunction}}^{\left(0,0,1\right)}\left[x,y,z\right]+\mathrm{dy}{\mathrm{SomeFunction}}^{\left(0,1,0\right)}\left[x,y,z\right]+\mathrm{dy}\mathrm{dz}{\mathrm{SomeFunction}}^{\left(0,1,1\right)}\left[x,y,z\right]+\mathrm{dx}{\mathrm{SomeFunction}}^{\left(1,0,0\right)}\left[x,y,z\right]+\mathrm{dx}\mathrm{dz}{\mathrm{SomeFunction}}^{\left(1,0,1\right)}\left[x,y,z\right]+\mathrm{dx}\mathrm{dy}{\mathrm{SomeFunction}}^{\left(1,1,0\right)}\left[x,y,z\right]+\mathrm{dx}\mathrm{dy}\mathrm{dz}{\mathrm{SomeFunction}}^{\left(1,1,1\right)}\left[x,y,z\right]$

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

In[105]:=

$\mathrm{dScalarFunction}=\mathrm{dScalarFunction}/.\left\{\mathrm{dx}\mathrm{dy}\to 0,\mathrm{dy}\mathrm{dz}\to 0,\mathrm{dx}\mathrm{dz}\to 0\right\}$

Out[105]=

$\mathrm{dz}{\mathrm{SomeFunction}}^{\left(0,0,1\right)}\left[x,y,z\right]+\mathrm{dy}{\mathrm{SomeFunction}}^{\left(0,1,0\right)}\left[x,y,z\right]+\mathrm{dx}{\mathrm{SomeFunction}}^{\left(1,0,0\right)}\left[x,y,z\right]$

$\mathrm{The}\mathrm{above}\mathrm{form}\mathrm{is}\mathrm{like}\mathrm{the}\mathrm{thermodynamic}\mathrm{expression}:\text{}\mathrm{dF}=\frac{\partial F}{\partial x}\mathrm{dx}+\frac{\partial F}{\partial y}\mathrm{dy}+\frac{\partial F}{\partial z}\mathrm{dz}$

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

In[106]:=

$\mathrm{CrazyFun}\left[\text{x\_},\text{y\_}\right]:=\frac{\mathrm{Sin}\left[5\pi x\right]\mathrm{Sin}\left[5\pi y\right]}{xy}+\frac{\mathrm{Sin}\left[5\pi \left(x-1\right)\right]\mathrm{Sin}\left[5\pi \left(y-1\right)\right]}{\left(x-1\right)\left(y-1\right)}$

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

In[107]:=

$\mathrm{theplot}=\mathrm{Plot3D}\left[\mathrm{CrazyFun}\left[x,y\right],\left\{x,0.1,\text{.9}\right\},\left\{y,0.1,\text{.9}\right\},\mathrm{PlotRange}\to \mathrm{All},\mathrm{Mesh}\to \mathrm{False}\right]$

Out[107]=

$⁃\mathrm{SurfaceGraphics}⁃$

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

In[108]:=

$\mathrm{Approxfunction}\left[\text{x\_},\text{y\_},\text{xo\_},\text{yo\_}\right]:=\mathrm{Series}\left[\mathrm{CrazyFun}\left[x,y\right],\left\{x,\mathrm{xo},2\right\},\left\{y,\mathrm{yo},2\right\}\right]//\mathrm{Normal}$

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

In[109]:=

$\mathrm{anapprox}=\mathrm{Plot3D}\left[\mathrm{Evaluate}\left[\mathrm{Approxfunction}\left[x,y,\text{.7},\text{.1}\right]\right],\left\{x,\text{.7}-\text{.1},\text{.7}+\text{.1}\right\},\left\{y,\text{.1}-\text{.1},\text{.1}+\text{.1}\right\}\right]$

Out[109]=

$⁃\mathrm{SurfaceGraphics}⁃$

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

In[110]:=

$\mathrm{Show}\left[\mathrm{anapprox},\mathrm{theplot}\right]$

Out[110]=

$⁃\mathrm{Graphics3D}⁃$

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]:=

$\mathrm{Table}\left[\left\{\mathrm{xo}\left[i\right]=\mathrm{Random}\left[\right],\mathrm{yo}\left[i\right]=\mathrm{Random}\left[\right]\right\},\left\{i,1,100\right\}\right];$

The next function automates the small approximating surface patch

In[112]:=

$\mathrm{ApproxPlot}\left[\text{i\_}\right]:=\mathrm{Plot3D}\left[\mathrm{Evaluate}\left[\mathrm{Approxfunction}\left[x,y,\mathrm{xo}\left[i\right],\mathrm{yo}\left[i\right]\right]\right],\left\{x,\mathrm{xo}\left[i\right]-\text{.1},\mathrm{xo}\left[i\right]+\text{.1}\right\},\left\{y,\mathrm{yo}\left[i\right]-\text{.1},\mathrm{yo}\left[i\right]+\text{.1}\right\},\mathrm{PlotPoints}\to 6,\mathrm{ColorFunction}\to \left(\mathrm{RGBColor}\left[\mathrm{xo}\left[i\right],\mathrm{yo}\left[i\right],\text{\#}\right]\&\right),\mathrm{DisplayFunction}\to \mathrm{Identity}\right]$

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]:=

$\mathrm{Clear}\left[\mathrm{GraphicsStack}\right]$

In[114]:=

$\mathrm{GraphicsStack}\left[0\right]=\mathrm{Show}\left[\mathrm{ApproxPlot}\left[1\right],\mathrm{DisplayFunction}\to \mathrm{Identity}\right]$

Out[114]=

$⁃\mathrm{SurfaceGraphics}⁃$

Here is the recursive function.

In[115]:=

$\mathrm{GraphicsStack}\left[\text{i\_}\right]:=\mathrm{GraphicsStack}\left[i\right]=\mathrm{Show}\left[\mathrm{GraphicsStack}\left[i-1\right],\mathrm{ApproxPlot}\left[i+1\right]\right]$

In[116]:=

$\mathrm{Show}\left[\mathrm{GraphicsArray}\left[\left\{\mathrm{GraphicsStack}\left[10\right],\mathrm{Show}\left[\mathrm{theplot},\mathrm{GraphicsStack}\left[10\right]\right]\right\}\right],\mathrm{DisplayFunction}\to \mathrm{\$DisplayFunction}\right]$

$\mathrm{Graphics}\text{::}\mathrm{realu}:\text{Argument in}\mathrm{RGBColor}\left[0.1947701615072844,0.1938584747649562,1.0083416833631114\right]\text{is not a real number between 0 and 1.}\mathrm{More\dots }$

$\mathrm{Graphics}\text{::}\mathrm{realu}:\text{Argument in}\mathrm{RGBColor}\left[0.1947701615072844,0.1938584747649562,1.0083416833631114\right]\text{is not a real number between 0 and 1.}\mathrm{More\dots }$

$\mathrm{Graphics}\text{::}\mathrm{realu}:\text{Argument in}\mathrm{RGBColor}\left[0.1947701615072844,0.1938584747649562,1.0083416833631114\right]\text{is not a real number between 0 and 1.}\mathrm{More\dots }$

$\mathrm{General}\text{::}\mathrm{stop}:\text{Further output of}\mathrm{Graphics}\text{::}''realu''\text{will be suppressed during this calculation.}\mathrm{More\dots }$

Out[116]=

$⁃\mathrm{GraphicsArray}⁃$

Created by Mathematica  (October 14, 2005)