I sometimes have trouble with very abstract areas of math. I like to have a mental picture of what I'm doing, and that gets hard when you do things at different/transformed levels. (Nyquist plots to solve PID control problems, for example.)
Something I'm good at is writing numerical solvers for various problems. I see an integral and think t=0; Int = 0; while(t<tend) {Int+=f(bla)*dt; t+=dt;}. Derivatives, gradients, laplacians, are similarly easy. I have a very literal idea of what is happening numerically. Searching solution spaces for maximums/minimums/intercepts is just a matter of applying an algorithm that evaluates points within the solution space.
I was wondering if mathematicians are good at generating these mental pictures, or if, at their level, it is all just abstract manipulation to them. How do you guys think of your problems?
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