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Re: Ohm's Law, Hooke's Law, etc.



Peter Vajk wrote:
*******************************************************
[SIMPLE PREMISE:] IF the functional relations mentioned above are
CONTINUOUS functions,

[SIMPLE CONSEQUENCE:] THEN, over SOME finite range of values, EITHER
relationship is very nearly LINEAR. (In calculus terms, every
continuous function, over some finite domain, may be described
arbitrarily accurately in terms of its value and its first derivative at
a point inside that finite domain.)

Jack Uretsky wrote in response:
*********************
Sorry, Peter, but continuity does not imply differentiability.
Mathemeticians give examples of functions that are everywhere continuous
(they
satisfy the axioms of continuity) and are nowhere differentiable.
Also, when material fractures, the strain is not a continuous
function
of the stress.
*********************

Hi all --
Jack is, of course, right -- I neglected to specify differentiability
as well as continuity -- albeit, in most simple physical, especially
classical mechanical. systems (short of quantum mechanics, chaotic
hydrodynamics, etc.) the two seem to go together.

With that correction, however, I stand by my earlier note. When
materials fracture, they have (clearly) exceeded the ELASTIC LIMIT
which I mentioned.

Best regards,

Peter Vajk
St. Joseph Notre Dame High School