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Calculus for mathematicians (1997) [pdf]

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Re: Calculus for mathematicians (1997) [pdf]

#31

Post by stephencanon »

impendia wrote: post #29
bandrami wrote: post #11

The hardest ( ie , best) math prof I ever had (I was in EE, he was in the math department) used to say that "engineers can teach calculus, but their students cannot go on to teach calculus". I'm impressed by someone who knows a subject deeply enough to take that long of a view. Personally, I just like the engineering view of dx and dy as simply being new variables (with caveats that we immediately forget). Which is w…

> Trig is a bunch of arbitrary formulas if you don't have the calculus behind them. Why do you claim this? You don't have to have seen calculus to appreciate how trig functions are defined, how to manipulate them, or how to use them in applications. As a math professor, I personally like the fact that we teach trig and exponential/logarithmic functions before calculus. They are (as you well know) exceedingly rich exa…

sin and cos are the natural basis of solutions of y'' + y = 0. How do you define them?

Re: Calculus for mathematicians (1997) [pdf]

#32

Post by btilly »

mafribe wrote: post #20
tel wrote: post #19

Earlier quoted context omitted.

I haven't read the paper, but if you define the reals as constructive Cauchy sequences you'd be forced into a coinnductive definition. Is that equivalent?

I'm not familiar with constructive Cauchy sequences. The usual way of using Cauchy sequences is to quotient them by the ideal of Cauchy sequences that converge to 0.

In constructivism, once you have a Cauchy sequence of rationals, that represents a real. However given two such Cauchy sequences, you cannot always prove whether one is less than, equal to, or greater than the other.

This has some interesting consequences. For example only continuous functions can be functions in constructivism. (If you try to construct a function that is discontinuous at a point, there are Cauchy sequences you can give it that you cannot assign to a Cauchy sequence coming out. So it is not a well-defined function.)

Re: Calculus for mathematicians (1997) [pdf]

#33

Post by tarentel »

stephencanon wrote: post #31
impendia wrote: post #29

Earlier quoted context omitted.

> Trig is a bunch of arbitrary formulas if you don't have the calculus behind them. Why do you claim this? You don't have to have seen calculus to appreciate how trig functions are defined, how to manipulate them, or how to use them in applications. As a math professor, I personally like the fact that we teach trig and exponential/logarithmic functions before calculus. They are (as you well know) exceedingly rich exa…

sin and cos are the natural basis of solutions of y'' + y = 0. How do you define them?

Ratios in a right triangle perhaps?

Re: Calculus for mathematicians (1997) [pdf]

#34

Post by ghshephard »

anon4 wrote: post #9
hudibras wrote: post #4

Definitely stashing this away in my time machine for when I travel back to the 17th century. Make both Leibniz and Newton cry...

If you're taking things like that back in time, make sure to translate them to French. Also, learn French. Not knowing French in the 17th century is like not knowing English today.

Also, learn French in Canada, not France prior to your trip back in time. I've been told that Canadians speak French like someone from the 1600s, and those visitors (in 2015) from France find it archaic to the ear.

Re: Calculus for mathematicians (1997) [pdf]

#35

Post by Tloewald »

bandrami wrote: post #11

The hardest ( ie , best) math prof I ever had (I was in EE, he was in the math department) used to say that "engineers can teach calculus, but their students cannot go on to teach calculus". I'm impressed by someone who knows a subject deeply enough to take that long of a view. Personally, I just like the engineering view of dx and dy as simply being new variables (with caveats that we immediately forget). Which is w…

Trig is the math of triangles and circles — it can be fully understood in geometric terms. Calculus requires far more foundation.

The linked article is interesting but the definition of differentiability looks wrong to me — maybe my brain needs more coffee but it looks like only linear functions are differentiable as defined.

I learned my calculus the pure math way — axioms and analysis. Epsilon delta arguments make more sense to me than "Ball". The fact that to clarify the examples the author resorts to epsilon delta description suggests to me that this approach is clever rather than clear.

Re: Calculus for mathematicians (1997) [pdf]

#36

Post by rndn »

bandrami wrote: post #11

The hardest ( ie , best) math prof I ever had (I was in EE, he was in the math department) used to say that "engineers can teach calculus, but their students cannot go on to teach calculus". I'm impressed by someone who knows a subject deeply enough to take that long of a view. Personally, I just like the engineering view of dx and dy as simply being new variables (with caveats that we immediately forget). Which is w…

> An integral is the inner product of a function and a suitably-dimensioned unit.

Does that mean that ∫f(x)dx is f.(dx, dx, …) = f(x_0)·dx + f(x_1)·dx + f(x_2)·dx + … for all x in the domain?

Re: Calculus for mathematicians (1997) [pdf]

#37

Post by johnaspden »

stephencanon wrote: post #31
impendia wrote: post #29

Earlier quoted context omitted.

> Trig is a bunch of arbitrary formulas if you don't have the calculus behind them. Why do you claim this? You don't have to have seen calculus to appreciate how trig functions are defined, how to manipulate them, or how to use them in applications. As a math professor, I personally like the fact that we teach trig and exponential/logarithmic functions before calculus. They are (as you well know) exceedingly rich exa…

sin and cos are the natural basis of solutions of y'' + y = 0. How do you define them?

coordinates of rotaty-thing

Re: Calculus for mathematicians (1997) [pdf]

#39

Post by monochromatic »

stephencanon wrote: post #31
impendia wrote: post #29

Earlier quoted context omitted.

> Trig is a bunch of arbitrary formulas if you don't have the calculus behind them. Why do you claim this? You don't have to have seen calculus to appreciate how trig functions are defined, how to manipulate them, or how to use them in applications. As a math professor, I personally like the fact that we teach trig and exponential/logarithmic functions before calculus. They are (as you well know) exceedingly rich exa…

sin and cos are the natural basis of solutions of y'' + y = 0. How do you define them?

Infinite sums are as clean as anything.

Re: Calculus for mathematicians (1997) [pdf]

#40

Post by threatofrain »

tarentel wrote: post #33
stephencanon wrote: post #31

Earlier quoted context omitted.

sin and cos are the natural basis of solutions of y'' + y = 0. How do you define them?

Ratios in a right triangle perhaps?

How did you get that information? How did you, in a non-magical way, go from information about an angle to information about a ratio?

If students don't know this information, then perhaps they are studying applications. So, what applications are students taught in typical trigonometric texts? Periodic behavior perhaps? Like sound? Only perhaps a brief blurb in the text that application is even possible. Perhaps they look at something about an incline plane. It is unlikely that they will touch projectiles.

It appears that trigonometry is there to give students some sense of mild comfort for future work in physics or engineering. This makes me think, "Why not statistics instead?"

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