# Plotting families (depending on integer parameter(s)) of curvilinear coordinate and isolevel lines and surfaces, resp. for functional and parametric surfaces, and for 3-variate scalar fields and volume deformations in 3D

To begin with a warning: I am a very experienced mathematician but a complete – 'greenhorn' – beginner in SAGE. I would like to be able to plot in the same 3D plot FAMILIES OF PARAMETRIC CURVES IN 3D (x(i)=x(t,i),y=y(t,i),z=z(t,i)) plotted via e.g. parametric_plot3D DEPENDING ON A NON-NEGATIVE INTEGER PARAMETER i in range(I+1) where the coordinate functions x,y,z are defined as functions of t by me and are not necessarily predefined SAGE functions, and where I can vary the non-negative integer I without varying the code. I would like to do the same for FAMILIES OF FUNCTIONAL SURFACES (z=f(x,y,i), DEPENDING ON A NON-NEGATIVE INTEGER PARAMETER i) using e.g. ParametricSurface, as well as TO DO THE SAME for FAMILIES OF PARAMETRIC SURFACES ( (fx(u,v,i),fy(u,v,i),fz(u,v,i)) ) via using e.g. parametric_plot3D DEPENDING ON A NON-NEGATIVE INTEGER PARAMETER i in range(I+1. I would also like to be able to do the same for FAMILIES OF IMPLICITLY DEFINED SURFACES IN 3D DEPENDING ON A NON-NEGATIVE INTEGER PARAMETER i, using e.g. implicit_plot3d ( (f(x,y,z)-C(i)==0, I in range (I+1)), C being a 1-dimensional array (in general algorithmic language) with entries – real numbers in the range of the values of the real-valued f), with the possibility of custom definition of f, I and C. Examining the SAGE tutorials and reference manuals, I have so far been able to find a model solution of this problem only for planar curves in 2D using plot() and only for SAGE-predefined curves (namely, polynomial curves): http://sage.maa.org/home/pub/140/

sage: lotsa_plots = sum([plot(x^n,(x,0,1),color=rainbow(5)[n]) for n in range(5)])
Already in this simple 2D case, if the SAGE-predefined function x^n in the above example be replaced by a custom-defined function f(x,n) , it is not clear to me (a greenhorn!) how to extend the above instance to plotting together the 2D graphs of f(x,n), color=rainbow(N)[n]) for n in range(N), where one should be able to vary N withouth modifying the SAGE code.
I think that providing examples solving the above tasks in the SAGE tutorials/manuals (or developing SAGE in this direction if the tasks are currently SAGE-unsolvable!) is of key importance FOR A GREAT MANY OF THE USERS (OR WANNABE USERS) OF SAGE WHO ARE EXPERIENCED THEORETICIANS BUT ARE NOT EXPERIENCED PROGRAMMERS.
To facilitate the eventual answering of these questions, I herewith propose a simple but comprehensive example. IF YOU PROVIDE COMPLETE ANSWERS CONCERNING THIS EXAMPLE, YOU WILL HAVE ANSWERED ALL OF THE ABOVE QUESTIONS TOGETHER. Moreover, your answer would certainly merit to be included among the examples in the next version(s) of the SAGE tutorial[(s) and/or reference manual(s).

Consider the 3D scalar field (argument is a 3D vector ...

It is not very easy to read the question. Would you mind editing using LaTeX syntax and indenting code blocks? ps Is this the longest question we have seen aruond ask.sagemath?