Differential Properties of Curves
Two examples of closed curves:
Here are two vectors with components given in terms of the parameter t:
In[68]:=
Both of these functions define a list of 3 positions along each axis in terms of a single parameter
In[69]:=
Defining Functions to Display Them
In[70]:=
In[71]:=
In[72]:=
In[73]:=
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]:=
Do the same thing for a different parameterized vector function:
In[75]:=
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]:=
Out[76]=
This is the arclength up to the parameter t
In[77]:=
Out[77]=
Applying this to the function Bendy defined above:
In[78]:=
Out[78]=
In[79]:=
Out[79]=
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]:=
Out[80]=
In[81]:=
Out[81]=
Alternatively, we can evaluate the expression for arc length numberically using the following:
In[82]:=
Out[82]=
In[83]:=
Example of a parameritized surface
In[84]:=
In[85]:=
Out[85]=
A Curve on a parameritized surface
In[86]:=
Out[87]=
In the above, I want to put the vine outside the surface so I scale it out a little bit...
Out[53]=
In[88]:=
Out[88]=
In[89]:=
Out[89]=
In[90]:=
Out[90]=
Multivariable Calculus: Mathematica Review
AScalarFunction is defined everywhere in (x,y,z)
In[91]:=
In[92]:=
Out[92]=
The following lines print and they define expressions.
In[93]:=
Out[93]=
Out[94]=
Out[95]=
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]:=
Out[96]=
Because it is now a function of w and v, the derivative with respect to x will vanish:
In[97]:=
Out[97]=
Two more flavors of derivatives, these are partial derivatives evaluated on the surface
In[98]:=
Out[98]=
In[99]:=
Out[99]=
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]:=
Out[100]=
The following is a derivative of the function along the curve parameterized by t
In[101]:=
Out[101]=
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]:=
Out[102]=
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]:=
Out[103]=
In[104]:=
Out[104]=
The next step eliminates second- and third-order terms… (remember, dx, dy, and dz are small)
In[105]:=
Out[105]=
Example of using Taylor Expansion to turn a function of two variables into an approximating function of four variables:
In[106]:=
Plot this function over suitable range of the variables x and y:
In[107]:=
Out[107]=
Now, approximate the function about a specific point (xo, yo), using Mathematica's Series function:
In[108]:=
and plot the approximate function in the neighborhood of (xo, yo):
In[109]:=
Out[109]=
Both the original and the approximate function can be plotted simultaneously:
In[110]:=
Out[110]=
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]:=
The next function automates the small approximating surface patch
In[112]:=
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]:=
In[114]:=
Out[114]=
Here is the recursive function.
In[115]:=
In[116]:=
Out[116]=