I am not sure the moderator will approve of this thread. It starts with a statement from Newton and then proceeds to build a model to test the statement, where the model to test the statement requires numerical integration in which I employed Mathematica. The results were somewhat unexpected and led to some further parametric modeling using different coordinate systems to build and evaluate the resulting integrals. I constructed spheres and cylinders to test the statement by Newton. The results were not fully consistent, in that the numerical evaluation of integrals disagreed at a certain level of precision. The results using spheres proved most variable under certain circumstances. The results using cylinders had no differences to very high levels of precision that would have shown that using the two coordinate systems would give different results. This then is an issue of when or how are the numerical results derived from using Mathematic noise or true variations in the calculated values.
This thread starts with a statement from Newton. It proceeds to a mathematical proof using Mathematica. This then leads to an issue of how reliable are the results of the numerical integration. In the spirit of full disclosure, I have been working with these models for several years and the comments from various members of the community have been very useful. In none of those threads was the real underlying issue discussed, though now that my analysis is finished, it is useful to broaden the discussion.
Any Wolfram Technologies discussions are fine to post if they follow the rules: http://wolfr.am/READ-1ST One of the rules is that you actually have to post your work with explanations and Mathematica code. It is needed to start the discussion and it would be a great read for many users here. If you need examples please take a look at the format of our Staff Picks section:
So please, EDIT your post and add Mathematica code. This all is explained in the guidelines http://wolfr.am/READ-1ST that we keep forwarding you to. Please make an effort and read them through.
Got it. Will try to do better in the future.
On another note, I have much of my work recorded in "formal" notes. Is it OK to attach these or even links to these to avoid directly posting what many would consider excessive detail...that's how I make my notes so that I can recall why and how I made certain decisions ... as well as why something turned out to be a dead end.