User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Exponent Combination Laws
I realized that the proofs of the Laws of Logarithms don't work according to the 2nd proof of Derivative of Natural Logarithm Function. Since I'm the one who added the second proof after the laws of logarithms were put up, I'm trying to add a proof for the laws that aren't circular. While Googling it I found someone who encountered the same problem on a homework assignment, and (lucky for me?) he didn't post his resolution. That is, he found a source for the the definition using $\exp$ as the inverse of the natural log, whereas he wanted to prove it using the definition


 * $\displaystyle \lim_{n \to \infty} \left({1 + \frac x n}\right)^n := \exp x$

I might wait until next beginning of next year when I do sequences and series and limits at infinity in detail. At the very least, I think I know enough to prove laws of exponents for integer indices, should be a simple proof by induction?, but I don't want to clutter PW with extraneous proofs. Thoughts? My semester just ended so I have some free time until Spring semester. --GFauxPas 12:09, 23 December 2011 (CST)


 * "Extraneous proofs": If it's a result which is interesting enough to document, then it's worth putting up on ProofWiki. One of the first questions you have when putting a site together like this is: "Is this statement too trivial to mention?" If you find yourself asking that question, then you will probably find yourself, sooner or later, making the wrong decision. Thus a vital step in a proof is glossed over, as its proof is "too trivial", and then a mistake creeps in because this point is insuffuciently understood. There's a biblical quote that applies here: something like "[The stone that the builders cast aside has become the keystone]." Hence my (own personal) decision that: there is no mathematical proof which is so simple and trivial it's not worth putting up here. I even believe we can put up a category of "examples" and/or "exercises" (but I haven't bent my brain cell in that direction yet).


 * As for the question about:
 * $\displaystyle \lim_{n \to \infty} \left({1 + \frac x n}\right)^n := \exp x$
 * Other way about, btw, it should be:
 * $\displaystyle \exp x := \lim_{n \to \infty} \left({1 + \frac x n}\right)^n$
 * (it means: "the $\exp x$ function is defined as being the limit of ..."


 * The aim is to prove that it follows strictly from the definition as given in these pages. Then we can confirm we have no circularity. --prime mover 16:35, 26 December 2011 (CST)
 * It is actually easy to prove that above definition coincides with the unique solution definition (the unique $f$ s.t. $f' = f, f(0)=1$) as, given uniform convergence (which is there on closed intervals), limits and differentiation may be swapped, yielding the desired immediately. I think it will be enough to just show equivalence of the definitions, after which all steps are allowed. Note that proving this equivalence of definitions as I sketched above builds on uniqueness theorems in differential equation theory and elementary calculus, so no dependency on logarithms required. The remaining equivalences I can't pull out of my hat at will.


 * P.m, I agree on your assessment of the exercises and such, compare the three example pages on Definition:Convex Set. Lastly, your quote originates from Psalms 118, verse 22. Also, this is quoted again in the New Testament. Just for completeness' sake. --Lord_Farin 16:57, 26 December 2011 (CST)
 * L_F: Examples: Yes, that page cited is an instance of how examples work - I may spend some time thinking about how to template-ize this. I'm on vacation all this week and my wife's at work, so (if I'm not spending the whole week watching DVDs she doesn't like) I may well think about this. On the other hand I've just got my hands on and, so, a-ha-ha, both are doubtful. --prime mover 17:18, 26 December 2011 (CST)

Arc Length
Let $y = f\left({x}\right)$ be real function such that $D_xf$ is continuous on the interval $\left[a..b\right]$.

The arc length $s$ of $f$ between $a$ and $b$ is defined as:


 * $s:= \displaystyle \int_a^b \sqrt{1 + \left({\frac {\mathrm dy}{\mathrm dx}}\right)^2}\ \mathrm d x$

Explanation
It makes sense to define the length of a line segment to be the distance between the two end points, as given by the Distance Formula:


 * $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$

Similarly, it is reasonable to assume that the actual length of the curve would be approximately equal to the sum of the lengths of each of the line segments, as shown in the diagram.

To calculate the sum of the length of these line segments, divide $\left[{a .. b}\right]$ into any number of closed subintervals of the form $\left[{x_{k-1} .. x_k}\right]$ where:


 * $a = x_0 < x_1 \cdots < x_{k-1} < x_k = b$

Define:


 * $\Delta x_i = x_i - x_{i-1}$


 * $\Delta y_i = y_i - y_{i-1}$

As the length of the $i$th line segment is $\sqrt{\left(\Delta x_i\right)^2 + \left(\Delta y_i\right)^2}$, the sum of all these line segments is given by:


 * $\displaystyle \sum_{i=1}^{k}\ \sqrt{\left(\Delta x_i\right)^2 + \left(\Delta y_i\right)^2}$

By hypothesis, $y$ is differentiable.

By differentiable function is continuous, $y$ is continuous.

Therefore we can apply the Mean Value Theorem on $y$. It follows that in every closed subinterval $\left[{x_{k-1} .. x_k}\right]$ there is some $c_i$ such that:


 * $D_xf\left({c_i}\right) = \dfrac {\Delta y_i} {\Delta x_i}$

Thus the approximate arc length is given by the Riemann Sum:


 * $s \approx \displaystyle \sum_{i=1}^{k}\ \sqrt{1 + \left(\frac{\mathrm dy}{\mathrm dx}\right)^2}\Delta x_i$

If there exist: a lower sum and upper sum such that:


 * A lower sum, $L \left({\left[{x_{k-1} .. x_k}\right]}\right)$ for each subdivision
 * An upper sum $U \left({\left[{x_{k-1} .. x_k}\right]}\right)$ for each subdivision

Then the exactly length of curve is defined by the definite integral:


 * $s:= \displaystyle \int_a^b \sqrt{1 + \left({\frac {\mathrm dy}{\mathrm dx}}\right)^2}\ \mathrm d x$