If f(x) is a polynomial, then its derivative f'(x) is the coefficient of t in the expansion of f(x+t):
[math]f(x+t)=f(x) + t \cdot f'(x) + \dots[/math]
things-that-blow-your-mind-thread
>>8124627
yes, the derivative is the linear part of the function
>>8124627
>what are taylor series expansions
>>8124627
how's freshman calc going, op
>>8124634
you can use the property in the op to define what the derivative of a polynomial should be in completely algebraic terms.
Another way would be to say that there exists a polynomial g(x,y) with
[math]f(y)-f(x)=(y-x)\cdot g(x,y)[/math]
Then f'(x) = g(x,x)
>>8124627
Fourier transforms
>>8125217
what does that have to do with anything?
The area under the curve of ln(x) from [1,e] is one. Woaooaooah iknowright.
>the unit circle has an area of exactly pi
what sorcery is this
>>8126933
It's a "things-that-blow-your-mind-thread," so maybe it blows his mind?
>>8127008
the unit circle technically has no area, as it is a 1 dimensional manifold
the unit ball or unit disk is what you're talking about
>>8127760
>one dimensional
it's not a line you idiot
>>8127833
fuck off
>>8127833
Found the teenager.
>>8127760
do circles not have area?
If you pick two 3D vectors at random, the chance of them being perpendicular is pretty much zero. However, for sufficiently high dimensional spaces (say, N=1million), it is almost guaranteed that two random vectors will be perpendicular.
This result was proven by Emmanuel Candes and Terry Tao in 2006 and has enormous implications for compressed sensing.
>>8127883
[eqn]\lim_{N\rightarrow\infty} \sum_{i=1}^N a_i b_i = 0 [/eqn]
It's a neat insight, but not terribly surprising when you think about it for a second; it's the law of large numbers.
Granted, while this is the important intuition of Compressive Sensing (CS), the more important results pertain to the Restricted Isometry Principle (RIP). It's funny though, extremely few projects that claim to use CS actually satisfy the mathematical conditions necessary to apply CS. Rather, the caviler application of CS is used more to hype grants than actually realize technology.
The big problem with CS is that it is often sold as a way to take fewer measurements when doing things like imaging, however time saved in measurement is completely paid back x10 in computation.
Still, CS is some cool shit. I've been working on a project for the past 2 years that integrates it into an RF imaging system.
My contribution to the thread:
Universality: The property that a function some where in it's domain approximates any other complex-differentiable function arbitrarily well.
https://en.wikipedia.org/wiki/Riemann_zeta_function#Universality
>>8127882
A circle is a 1-sphere. A disk is a 2-ball. Spheres are hollow. Balls are solid.
>>8127882
> do circles not have area?
No. Disks have area, circles have length.
circle: x^2+y^2=r^2
disk: x^2+y^2<=r^2
Similarly, balls have volume, spheres have area.
sphere: x^2+y^2+z^2=r^2
ball: x^2+y^2+z^2<=r^2
>>8127833
A mapping from R^M->R^N has dimension M, not N (i.e. it's determined by the structure, not the space in which it's embedded).
A closed curve that does not intersect itself divides the plane into 2 regions. This just blows my mind.
>>8127883
>the chance
Lrn2probabilly
>>8128007
the proof blows my mind
Here's the one that's been bothering me for a while
Any continuous function that maps a compact(bounded and closed) convex set to itself will always have a stationary point i.e. f(x*) = x*
>>8128212
Seems pretty intuitive desu.
>>8127883
>This result was proven by Emmanuel Candes and Terry Tao in 2006
This "result" is obvious and has been known for centuries. Candes and Tao's work is important, but your statement is an absurd description of their work.
>>8127960
One of the best posts I've read on /sci/ for a long time.
I still don't understand why the antiderivative and the area under a curve are related.
Vector calculus is neato. Greens and stokes theorems namely
>>8128573
Imagine you integrate a derivative. What do you expect if you integrate an infinitesimally small range (1 to 1.000...001)
>>8128359
Straight from wiki
The theorem has several "real world" illustrations. Here are some examples.
1. Take two sheets of graph paper of equal size with coordinate systems on them, lay one flat on the table and crumple up (without ripping or tearing) the other one and place it, in any fashion, on top of the first so that the crumpled paper does not reach outside the flat one. There will then be at least one point of the crumpled sheet that lies directly above its corresponding point (i.e. the point with the same coordinates) of the flat sheet. This is a consequence of the n = 2 case of Brouwer's theorem applied to the continuous map that assigns to the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet immediately beneath it.
2. Take an ordinary map of a country, and suppose that that map is laid out on a table inside that country. There will always be a "You are Here" point on the map which represents that same point in the country.
3. In three dimensions the consequence of the Brouwer fixed-point theorem is that, no matter how much you stir a cocktail in a glass, when the liquid has come to rest some point in the liquid will end up in exactly the same place in the glass as before you took any action, assuming that the final position of each point is a continuous function of its original position, and that the liquid after stirring is contained within the space originally taken up by it.
>>8128573
you need to review the fundamental theorem of calculus then. it's a big result, you might want to learn some rigor if you really want to understand it.
>>8124627
The fact that f(x+t) = f(x) + t*f'(x) for small t is the definition of the derivative...
