More Writing: Calculus and Area/Volume Calculations of Arbitrary Shapes

 

I re-read some of my blog posts, and I think I should still be "somewhat occupied with writing," as I had originally said.  I should plan more essays and write them out.


Here is an idea I feel comfortable blurting out today.  If you have ever wanted to calculate the area of some shape, let's just say in 2-space for simplicity, you might have wondered what mathematical tools you could use to do that.


The main tool is integral calculus, which calculates the area under a curve.  You can calculate areas between two different curves, too, which means in general, if you have an enclosed shape, calculus can help you calculate the area of that shape.


I claim, without proof at this time, that any shape that doesn't have any curves can be captured with triangle-components alone.  If you have a pentagon, or a straight line, etc., you can include parts of triangles to construct the shape.  So that part is easy, even though, e.g., the triangle formed by the line f(x) = |x| is not differentiable.  Using piecewise functions, we can surely calculate the area in between two curves when the curves have nothing to them other than straight lines.


The challenge is curves, though.  You might say, "We can just use polynomials to capture the different curves."  I claim that that might not work, because there are uncountably infinitely many Taylor series.  Of course we have regular polynomials of finite degree, and trigonometric curves, like sin and cos curves.  At the same time, I believe that there are uncountably infinitely many curves beyond that.


Perhaps we could restrict attention to Taylor series that we could write down with some formula; there are only countably infinitely many of these.  It might be the case that the set of all "drawable" curves, in the physical universe, must also be countable.  How could we light up even the pixels of the screen in a way that captures truly "uncountable curvature"?


Another interesting facet of this is, we might be able to explore meaningful functions, beyond the trigonometric functions, using Taylor series.  In particular, any time we can conceive of a curve that is curvy and not angled that violates the rule of "admitting only finite-degree-polynomial or trigonometric curves", we can consider the possibility of a new type of function that we might wish to study.  In general, one great way to explain new functions is to write out power series functions for curves that we know don't conform to finite degree polynomials or trigonometric functions.  Upon studying these functions, we might be able to develop a treasure trove of new ideas for functions--and what they might mean, if we pretend that we are discovering a new function the way mathematicians decades ago discovered the trigonometric functions by thinking about triangles.


Ultimately, this is a cool research project for anyone who wants to try to find something new and not necessarily all that useful in basic, undergraduate-level mathematics.  I hope you have fun with it if you try it!  (I might try it myself someday, but not right now.)

Comments

Post a Comment

Popular posts from this blog

Math Update: My P != NP Proof Might Be Correct!!!!! -- (update, no it's not, see the note at the beginning and the updates near the bottom)

EDIT:  No, this proof is not correct.  See the updates at the bottom for details on why (based on Dr. Zelinsky's comment).  Fortunately, my GLL proof and MSR proofs still appear to be correct. It could be that Joshua Zelinsky's comment actually helps me to complete the proof. The GLL proof appears to be correct, AFAIK. The thing is, the fact that we can prove "PA is consistent --> P != NP" within PA and the fact that we can prove "PA is consistent" as a theorem of ZFC just shows that P != NP is a theorem of PA.  That isn't a contradiction, it's just another step in the proof. I think the striking idea is, P != NP is a theorem of PA, but not of ZFC.  That is what I worked to prove.  AFAIK it is possible for a proposition to be a theorem of PA but not of ZFC.  In particular, PA can be proved consistent from ZFC, so we would think it s possible and even likely that there are some non-theorems of ZFC that are theorems of PA. So this is great!  It l...
Read more

Hello, Texas - A Quick Monopoly Lesson

  So today, I thought I saw some people running around blurting code gestures at me.  One of the codes seemed to say, "Riot for a[buse] v[ictim] time -M".  Oh, M?  M as in Monolith, or M as in Monopoly fan who doesn't get it? Here, Houstonian code-talkers...let me teach you how to play Monopoly, since you are so hungry for free knowledge:  - Make deals with other players that benefit you.  - Offer deals to difficult players that are somewhat beneficial to them, knowing that they might reject.  - Watch which deals other players reject, and identify what *deal types* they are not likely to accept from you or anyone else.  - Be very very careful about your own deals--and behavior--lest you get figured out by your opponents and backed into a corner, with good, beneficial deals being made without you by cunning adversaries who know your every move. There's a poker lesson in there, too, but I don't want to get too educational right now.
Read more

Flash Non-Fiction: James Bond and Masculinity

  This was going to be a longer essay, but I decided to just blurt out my point rather than trying to write well. Basically, I think James Bond movies are good, and the songs are high-quality too, but it sends semi-dangerous messages to men in particular about what it means to be a man.  The Bond character is intransigent, aggressive, violent, and absolutely sure of himself in all cases.  He is portrayed as a big winner--able to date and have sex with beautiful women, almost impossible to kill, and lethal and elite at both armed and unarmed combat. I'm concerned that some men, including my estranged father ASW, have had their sense of morality warped by Bond movies.  They think that this dramatically presented way to be masculine is the only way.  I think that's unhealthy, and when people get the not-so-original idea to "try to be more Bond-like" in their personal lives, that leads to catastrophic conflicts with and major potential for deep pain for people that ...
Read more