derivative, nutshell:
- algebraic: just change of coordinates from value domain to difference domain
(which is best understood via discrete differences)
- analytic: Taylor: 0th order / 1st order / etc. approx
[What is actually taught? The Taylor POV:
"zoom in on a function and it looks like a line"
BTW, good use of calculator; can even do higher derivs (if kill linear term; look at cos)!]
- higher dimensions
no longer "deriv of function is function": it's a differential form
(dually, vector field)
[interestingly, integrals in higher dimensions are easier than derivs,
b/c there's no new concept (just visualize higher dim): just chop domain
higher dim'l derivs do require tricky concept of differential form]
integral, nutshell:
- take measure theory POV, with lower & upper:
just bound below / bound above
(equiv, approximate *function*)
- works in higher dimensions too
- my notation
- infinities tricky (absolute convergence)
- discrete POV: summing series
- double/triple integrals are *computational tools*
- can also integrate w/r/t *orientation*
- other POV is to choose some point: this doesn't bound
but does approximate, and can be made very good