- #26

mathwonk

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TA's can certainly teach elementary calc AT leAST IN THE WAY THEY leARNED IT, BUT a prof can teach it anyway he/she wants to.

there are many many ways to explain a subject you really understand, and have spent years thinking about.

i can teach calc with insights from archimedes, or newton, or fermat, or euler, riemann, or errett bishop, john tate, or any of the many other masters i have come to know, either personally or through their writings.

i may be wrong, but i think i understand calculus better than most textbook authors, [not the ones i recommend here, but many of the ones we often use], and I try to incorporate my understanding in every class.

how many elementary calc profs will show you how to compute the volume of a 4 dimensional sphere by the same technique used for a three dimensional one?

see if you can read the treatment in a standard book and see how to extend it. many of them give it in a way that is not clearly extendable. why?

how many know that newton already proved "riemann integrability" of monotone functions? (see michael comenetz's calc book). or that riemann himself already proved that a riemann integrable fucntion is equivalent to one whose discon tinuities have "lebesgue measure" zero. (see riemanns works, paper on trigonometric functions). or that fermats algebraic method already computes the derivative of every function in elementary calc, without limits, except sin and exp.

of course many profesors know these things, but not all, and probably not most TA's. I have even seen [and reviewed] calc books, good ones too, whose authors were clearly unAWARE OF these points, since they contradicted them.

how many TA's know the connection between differential forms AND DERHAM COHOMOLOGY? IT IS HARD TO GIVE insights into more advanced material unlessone knows it oneself. when i taught advanced calc the first time, i used stokes theorem to rpove the non existence of vector fields on an (even dimensional) sphere, and the brouwer fixed point theorem. and this was in 1971.

there are many many ways to explain a subject you really understand, and have spent years thinking about.

i can teach calc with insights from archimedes, or newton, or fermat, or euler, riemann, or errett bishop, john tate, or any of the many other masters i have come to know, either personally or through their writings.

i may be wrong, but i think i understand calculus better than most textbook authors, [not the ones i recommend here, but many of the ones we often use], and I try to incorporate my understanding in every class.

how many elementary calc profs will show you how to compute the volume of a 4 dimensional sphere by the same technique used for a three dimensional one?

see if you can read the treatment in a standard book and see how to extend it. many of them give it in a way that is not clearly extendable. why?

how many know that newton already proved "riemann integrability" of monotone functions? (see michael comenetz's calc book). or that riemann himself already proved that a riemann integrable fucntion is equivalent to one whose discon tinuities have "lebesgue measure" zero. (see riemanns works, paper on trigonometric functions). or that fermats algebraic method already computes the derivative of every function in elementary calc, without limits, except sin and exp.

of course many profesors know these things, but not all, and probably not most TA's. I have even seen [and reviewed] calc books, good ones too, whose authors were clearly unAWARE OF these points, since they contradicted them.

how many TA's know the connection between differential forms AND DERHAM COHOMOLOGY? IT IS HARD TO GIVE insights into more advanced material unlessone knows it oneself. when i taught advanced calc the first time, i used stokes theorem to rpove the non existence of vector fields on an (even dimensional) sphere, and the brouwer fixed point theorem. and this was in 1971.

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