Thread: What are Linear approximations used for

Originally Posted by ThePerfectHacker

What JakeD said about the MVT is exactly what I meant when inequalities are used non-stop in number theory. However, when in physics we drop a term because we assume it does not exist (since it is almost zero) we need to consider in math, which is what the error terms accomplish.

(Evil laugh) However in the 50's in Quantum Field Theory terms in the perturbation expansion were sometimes ignored because they were infinite...

I seem to recall a number of topological proofs (based on the series definition of continuity) that depend upon dropping small terms as well.

And aren't affine spaces based on linear approximations? I'm not sure about this one because I don't have much experience with affine spaces. Something about them being a "flat" approximation to a curved space.

-Dan

Originally Posted by topsquark

I seem to recall a number of topological proofs (based on the series definition of continuity) that depend upon dropping small terms as well.

I do not know topology (noob ) but I am sure if those "drop of terms" is proved. So it is okay.

Here is what I am reffering to when I say approximation,
It is known that a swinging pendulum satisfies (am I right?):
sin y+y'=f(t)
What physcisits do is approximate the solution for small t that is sin t=t then,
y'+y=f(t)
This is a linear differencial equation and it becomes simpler.
However, a mathematicians would not be satisfied with such an approach because it is not exact that t=sin t
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To add I think there is something called "linearizing" in applied math to simplify the functions but mathematicians do not deal with such problems. Even more they are useless.