I was wondering what would be the best way to make something like what is shown in the attached file using Adobe Illustrator? Any suggestions? Or another Adobe App would be more suitable?
Doable, but a lot of work. The simpler answer to this therefore is to use MathLab/ Mathematica right away or a 3D program and export it from there as an SVG or whatever. Otherwise you can of course trace over your reference images and mangle the curves in whatever manner you fancy, it's just going to be laborious and tedious and figuring out the overlaps and coloring of e.g. a gradient mesh will be just as teeth-grindingly annoying.
Appreciate your input. I fugured myself that this would be exceedingly difficult but I wanted to cross-check with someone else more experience than me.