Derivatives

[K^2]

I'm still with KvickFlygarn87 on that. They're weird. Anything to do with SO(N) is weird. Trig functions are just a special case of that weirdness.

[Doozler]

I never got comfortable enough with groups to know what is weird and what is not! I'd enjoy reading your ruminations on it, but I'm not sure its appropriate for a thread on 8th-grade derivatives

Maybe we need a thread-lecture of group theory...

[ZetaX]

K^2: This is not an effect of trig functions but of exponential ones (as sin, cos etc. are expressible by them). The usual example is e^(-1/x^2) for similiar effects. This should not have much to do with special properties of SO(IR^n), and the weirdness vanishes completely if you change to complex calculus.

Edited April 4, 2014 by ZetaX

[KvickFlygarn87]

Hm. Maybe the derivative of cos(x) is sin(x)? I used a calculator to derive a few numbers and it does seem like it.

[Doozler]

You got it!

Now work out the derivative of Sin(x) and we'll tell you how pendulums work!

[Red Iron Crown]

Um, the derivative of cos(x) is -sin(x).

[KvickFlygarn87]

I'm thinking it may be either tan(x) or cos(x), probably cos(x).

Math in progress!

EDIT: Math completed, the derivative os sin(x) is cos(x)!

Edited April 4, 2014 by KvickFlygarn87

[Themohawkninja]

Derivatives in terms of mathematics is the slope of the tangent line (i.e. what the slope of the curve at a given point on the curve appears to be). When applied, this becomes the rate of change of something.

Here's the example I like to use. Let's say we have a graph that shows an objects' position over a period of time:

• The first derivative is velocity (the rate of change of position over time)
  
• The second derivative is acceleration (the rate of change of velocity over time)
  

There are applications for derivatives above two, but they get much more abstract and less applicable as I understand it.

[Doozler]

Um, the derivative of cos(x) is -sin(x).

Doh!

Math completed, the derivative os sin(x) is cos(x)!

Good work. Next stage is pendulums and springs!

Pull down a weight on a spring (or push a pendulum away from vertical) and there is a force pulling it back. The force gets bigger the further you pull it, and it pulls backwards (in the opposite direction to your pull). So the force on the weight for a given position x is:

F(x) = - k x

Where k is just a number telling you how beefy the spring is. Let's pretend it equals 1 for now. Your tasks:

• Write the acceleration of the weight for a given position.
  
• Re-write the acceleration in terms of derivatives of x (Themo gave you the answer)
  
• Guess a function x(t) that fits the equation
  
• Check that it does!
  

[KvickFlygarn87]

Doozler: Thanks for giving me something solid to work with! I learn best if I get to step up slowly. Although, I'm on my phone right now, I'll do that either later or tomorrow

[K^2]

K^2: This is not an effect of trig functions but of exponential ones (as sin, cos etc. are expressible by them). The usual example is e^(-1/x^2) for similiar effects. This should not have much to do with special properties of SO(IR^n), and the weirdness vanishes completely if you change to complex calculus.

The SO(2) is isomorphic to U(1), which is an imaginary exponent. So yeah, of course I can write everything down as a linear combination of exp(ix) terms, but that doesn't make it any less weird, or that weirdness any less relevant to structure of SO(2).

But SO(2)/U(1) is about as simple as non-trivial Lie Groups get, I'll give you that. At least, SO(2) is Abelian. All of my work is in SU(3)xSO(1,3). SU(3) is non Abelian, and has 8 generators. Eight. That means I have eight self-interacting gauge fields to deal with. Particle physics is really, really weird. And while they rarely show up directly, this weirdness is closely related to weirdness of trig functions. And hyper trig functions. Hey, at least my SO(1,3) is global. Stochasty deals with local SO(1,3) on top of all of the above, if I recall correctly.

I think, my main point is that there is a very deep hole you are staring down when you start asking questions about derivatives. But you can take it a step at a time with some guidance.

[Spaceradsoc]

Here's a book for you, you'll probably use it later anyway:

http://www.bokus.com/bok/9789144067650/analys-i-en-variabel/

It is not very good on its own as a starter book, you need some kind of lectures to combine it with. However, the structure is good and fills gaps.

[KvickFlygarn87]

Hmm. Spaceradoc, according to the comments it's bzd for the unknowing reader. I'm taking unknowing to a whole new level :-/

[Doozler]

This may sound insulting, but if you want further reading have you looked at

http://www.dummies.com/how-to/education-languages/math/Calculus.html

It has an intuitive approach (talks about slopes of hills etc) and more importantly is online and therefore free!

[Spaceradsoc]

It takes some time to get used to, in the beginning it may be hard, but after a while you will grow to like it. I'm not sure there are any options to that book anyway.

Maybe the fault with some of the people rating it is that they don't do the math themselves while reading it.