

Ask HN: A calculus theorem question?  - zeynel1

Any curve can be considered to be made of sufficiently small straight lines. http://musr.physics.ubc.ca/~jess/hr/skept/Math/node10.html<p>What is the name of the theorem which states this fact? I know integral calculus sums small units of the curve; but what is the name of the theorem?<p>I asked the same in mathoverflow but it has been downvoted and got no answer. http://mathoverflow.net/users/9811/zeynel<p>Thank you
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drallison
The problem is that your assertion is not true for any arbitrary function. For
example, consider a function(aka curve) defined by: f(x) is x if x is
irrational and 1 if x is rational. This function is discontinuous and cannot
be approximated by shorter and shorter straight lines.

The calculus concept you are reaching for is continuity. If f(x) is
continuous, then for any e>0 there exists a d>0 such that abs( f(x) - f(y) ) <
d if abs( x - y) < e. That's the famous epsilon-delta mantra of the calculus.
See
[http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_...](http://en.wikipedia.org/wiki/\(%CE%B5,_%CE%B4\)-definition_of_limit).

Continuity is a powerful property that enables much of what is useful in the
calculus. Continuity of a function and its derivative is even stronger;
continuity of the function and first and second derivatives is stronger still.

~~~
zeynel1
"Consider a function(aka curve)..."

Thank you for this answer and I apologize for the ambiguity in my question,
especially, regarding the word "curve."

By "curve" I meant "curved line" or even more precisely, a "curved line
without a gap." Obviously this is an assumption; I ignore if the line is made
of points; because I want to compare a curved line with a straight line. So I
believe that the question is not about "continuity."

(By the way Galileo has a nice discussion about lines and points here
[http://books.google.com/books?id=SPhnaiERbWcC&dq=galileo...](http://books.google.com/books?id=SPhnaiERbWcC&dq=galileo%20galilei&pg=PA26#v=onepage&q&f=false))

Maybe I can ask the same question like this. Here are 3 numbers: 0.0872664626;
0.1047197551; 0.1221730476. Is it possible to tell if these are points on a
straight line or if they are points on a curved line? I believe both;
depending on the scale they are plotted; the numbers would look like a
straight line on some scale; but they are points on a sin curve. In other
words, nature does not distinguish between straight line and curved line; the
same three numbers can be read both ways.

So I am looking for a theorem that would describe this fundamental property of
nature.

Thanks.

~~~
drallison
For x sufficiently small, sin(x) looks like x (x in radians). That's because
if you write the Taylor's series expansion of sin(x) and ignore high order
terms, you get x. The property you are looking for is continuity.

------
rcfox
It's not really a theorem; it's the definition of the derivative.

f`(a) = lim h->0 (f(a+h) - f(a))/h

~~~
zeynel1
Thank you!

Is it correct to say that the definition of the derivative states the
equivalence of the secant and tangent lines?

I am looking for a theorem that states the equivalence of curved and straight
lines. Does this make sense?

~~~
rcfox
Well, the tangent and the secant will converge to the same thing as h -> 0.

I'm not sure what you mean by "the equivalence of curved and straight lines"
though.

~~~
zeynel1
Tangent and secant converge to the same [point] but as far as I understand
this is not a statement about the equivalence of the curved and straight
lines. By this I mean something like Archimedes's method of exhaustion
[http://en.wikipedia.org/wiki/Method_of_exhaustion#Use_by_Arc...](http://en.wikipedia.org/wiki/Method_of_exhaustion#Use_by_Archimedes)
when he approximated a circle (curved line) with a polygon (straight line).

I would think that there would be a theorem in mathematics generalizing this
fact; something like; "curved line is equivalent to straight line." It depends
on how we divide (or scale) a line that gives it its "curvature." Or "there is
no curve but curvature." Something like that?

~~~
rcfox
Around a given operating point, you can approximate any function with a
polynomial (<http://en.wikipedia.org/wiki/Taylor_series>). This includes
approximating a polynomial with a lower-order polynomial. A line is first-
order polynomial. Therefore, around a given operating point, a line can
approximate a function.

The key phrase there is "around a given operating point". If you zoom in close
enough, a line and an exponential will appear to be the same. Outside of this
area, they will clearly diverge.

So you can't really say that a curved line is equivalent to a straight line,
but you can say that sin(x) = x, for small values of x (and tolerable levels
of error in sin(x)). In fact, that comes up quite often in engineering.

~~~
zeynel1
"you can say that sin(x) = x"

Yes, this is the small angle approximation
<http://en.wikipedia.org/wiki/Small-angle_approximation> and also the example
I gave above: 3 points on a sin curve may be considered a straight line or a
curved line; 1 meter on earth's orbit is a straight line. The same idea comes
up in the string theory too as a pun between point and line.

I am surprised that there is no mathematical theorem to express this
fundamental property of nature. Or maybe I am not expressing myself clearly.

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eiji
In mathematics you always try to minimize the preconditions, and the "theorem"
you are looking for is likely a combination of strong special proofs.

Maybe <http://en.wikipedia.org/wiki/Trapezoidal_rule> comes closest to what
you are looking for. It states that you can approximate the definite integral
using "straight lines", which includes your case, however, this rule has
preconditions for the curve. HTH

~~~
zeynel1
Thanks. This looks exactly like an "application" of my putative theorem. The
trapezoidal theorem assumes that straight line and curved line are the same
thing with different curvature (I think); the way a circle and ellipse are the
same figure but have different eccentricity.

------
zeynel1
Some answers in stackexchange too:
[http://math.stackexchange.com/questions/6136/is-there-a-
math...](http://math.stackexchange.com/questions/6136/is-there-a-mathematical-
theorem-that-states-the-equivalence-of-curved-and-straigh)

