I noticed that the chapters in my math and physic book tend to include a few review questions that require ingenuity in working with an introduced concept. I believe these questions are key to furthering an understanding of an established idea.
Whenever I try to apply concepts learned from my math and physic books, I have thought experiments but I would always run into trouble because I would ask questions that require esoteric information.
For example: After learning about sectors, circumference and all the basic formula about circle I figured I could use the equation I knew to find the volume of a sphere. I figured that if I found the 90 degree sector of a circle with radius x, I could multiply that by the circumference of a circle to get the volume of half a sphere. (Mapping the vectors along the perimeter of a circle around the center point will give me a solid.) I learned later that this doesn't work because of another theory that involved something I'd never even heard before. (Integrals; http://www.physicsforums.com/showthread.php?t=63654)
Things get more complicated when I start to use real objects. I feel like I don't know enough to do problems. Is this just because I am not gifted with the ingenuity to manipulate basic arithmetic to find the answer to answer something complex? (In other words, I don't have the capability to break down a system into feasible components.) Or are some concepts really esoteric?
I know what I'm asking here: How does the genius that came up with the correct formula come about to create the formula.
I pray it isn't hereditary.