Re: The logical structure of calculus, request for help.

On Wed, 04 May 2005 23:36:48 -0400, "A. Boom" <aboom@xxxxxxxxxx>
wrote:

>Consider calculus of a real variable.
>
>Is the following the logical structure for at least differential
>calculus, assuming knowledge of algebra, functions, set theory?
>
>* minimal knowledge of vector spaces.
>- Definition of a vector space.
>- Definition of a subspace.
>- Theorems involving subspace.
>- Definition of a norm.
>

I would say none of the above are neither prerequisite to nor normally
covered in the typical differential calculus course.

>* minimal knowledge of differential calculus.
>- Definition of a limit.
>- Theorems involving combinations of limits.
>- Definition of continuity at a point.
>- Definition of continuity on an interval.
>- Theorem: The intermediate value.
>- Theorems involving combinations of continuous functions.
>- Definition of derivative at a point.
>- Definition of derivative on an interval.
>- Theorem that differentiable at b implies continuous at b.
>- Theorems for computing derivatives; power law, etc.
>- Theorem of the chain rule.
>
>
>The above should be the bare minimum and a logical structure for
>teaching calculus?
>
>Note, I've left out definitions and derivatives of various functions,
>including the trigometric and exponential, theorems and definition
>(extreme value) used in max/min and optimization problems.
>

Why leave these and other applications out? Without working
applications, I think most students would find calculus a pointless
and sterile exercise in abstraction. Your description generally looks
like what a graduate student, with no teaching experience, who has
just been through a good linear algebra and advanced calculus course
might design to show his newly learned wonders of mathematics to all
those inquiring freshman minds.

--Lynn
.

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