First post

This blog is dedicated to all things mathematical. Rather than inflict my current (miniscule) readership with entries about things that they most likely don't find particularly interesting, I've decided to create a second blog with math as its primary orientation. It is my sincere hope that people with like minds will find my entries, and (just as important) the comments about them from others, to be interesting and engaging.

<"...and now a word from our lawyers..."> This is a work in progress. This is the work (presently at least) of a STUDENT of math. It is NOT holy writ. Therefore it may, from time to time, contain errors. You are strongly encouraged to verify any and all items discussed here with other sources (like your textbook). I don't want the following to happen:

"Director, I'm sorry, but the probe to Alpha Centauri is no longer responding. We think that it blew up."

"This is terrible! Do we know what caused the problem?"

"Well.... our investigation showed that programming for some critical systems were based on equations taken from the blog of a certain "sevej." These equations turned out to be wrong."

"Curse that sevej, and curse his blog!"

You get the idea.

That said, I'd like to think that this forum will provide a place where math can be discussed freely and openly. Therefore, if you think you've found an error, or you don't like the way something is explained, submit a comment! One of three things will happen; either your, or my, or perhaps both of our thoughts on the subject will be challenged, and this will result in our collective understanding being enhanced as it becomes more closely aligned with reality. In all cases, I promise to review your comments and correct these flawed entries or (better still) submit additional entries that explain how the original error was made and show the correction. I like to think that this will be of assistance in preventing myself and others from making similar mistakes in the future.

Most importantly, I'd really really like to hear your thoughts about math. Submit comments (or an e-mail if you are more comfortable), about what you are learning and what you find interesting. Every individual makes discoveries as they study, and each of these discoveries is unique and valuable. It matters not, that Euler or Gauss figured out something similar two or three hundred years ago; your discovery is just as wonderful because you made it. I like to think that once you experience that "Eureka" feeling that Archimedes of ancient Greece is said to have felt, you won't be able to live without it. You will continue to make more and more discoveries on your own, about things that less and less people have thought about, so that, before you know it, people will be reading about your theorems in their math textbooks!

Now, of course this is a safe place and "kid friendly". Question (notice I didn't say "attack") the idea, not the person submitting the idea. But please do question.

Lastly, I'd like to note that I am presently studying "the Calculus." Please do not find this "off-putting." I will happily discuss any and all math subjects up to the limits of my knowledge. Have a question about how long-division works, send it in. Don't understand multiplication of binomials, send it in. Can't figure out elliptical functions, send it in (we'll figure it out together 'cuz don't understand them either - yet.)

As mentioned elsewhere, I am currently a Mathematics Major at California State University at San Bernardino. I've just completed the first week of Calculus 212 which is the 2nd course of Calculus, which is slated to consider integration.

During our first class period we looked at indefinite integration as the opposite of differentiation, i.e. that the indefinite integral was equivalent to the "anti-derivative" (sevej watches as the eyes of all of the non-mathies glaze over). Anyway, one of the key concepts was that integration is "linear" (this is how it is described here in the section entitled "Linearity" concerning summations.)


This "linear" property allows one to take an integral apart at the +/- signs and handle each term separately.

We also discussed the formulas for constant acceleration, which are:

and

I remember studying these equations in Physics class in High School. Their derivation was not discussed (appropriately so, as we "only" had High School Algebra under our belts at the time), although I remember Mr. Lew hinting that something interesting was going on in the way that one equation was connected to the next. This "interesting" thing turns out to be differentiation, and with modicum of knowledge about how to take a derivative, one can see that v(t) is indeed the first derivative of s(t).

Oh, that reminds me, I should probably explain all of the symbols I've used (silly me):

s(t) = distance as a function of time
t = time
a = acceleration (constant acceleration in this case)
v₀ = initial or starting velocity
s₀ = initial or starting distance
v(t) = velocity as a function of time
a(t) = acceleration as a function of time

Well, in my present consideration, we considered these equations from the "bottom-up" so to speak. Starting with the equation for acceleration,



we take the indefinite integral (with respect to "t") to arrive at


The "constant of integration" C in this case can be interpreted as our initial velocity (v₀), to that our equation becomes


In like manner, if we integrate this equation, it becomes


Here, the "constant of integration" C is interpreted as the initial distance (s₀), so that it becomes

and, we're right back where we started from.