/sci/ - Science & Math

It isn't.

>babby's first calc 2

>>3985081

Let me rephrase for you.

<span class="math">\displaystyle \frac{d}{dx}[e^x] = e^x[/spoiler]

<span class="math">\displaystyle \int e^xdx = e^x + C[/spoiler]

WHY

>>3985093
imaginary numbers
thats why

The derivative of e is zero.

The derivative of e^x is the original function e^x.

>>3985089

You cover e in calc 1. I'm in calculus 3. In fact...I wrote this lovely limit proof of e a while back.

I just haven't given a shit about e till now. IT THINKS IT'S SO SPECIAL.

>>3985104

I don't think I would call that a proof of e.

>>3985104
um, not a proof.

>>3985104

Try harder.

You assumed definitions of e and of the log natural function. That's like saying "The bible is true because it says so in the bible."

Try the infinite sum (0 to inf) of 1/n!

So would you call it the limit definition of e?

I just did some quick searching and found a neat power series representation of e which proves its interesting differentiation pattern.

But so far none of you have explained the implications that this has in the physical world.

If I have a parabolic function modeling position, I'm going to have a linear line modeling velocity and a horizontal line modeling acceleration.

If I have an exponential function modeling position (generalized to <span class="math">\displaystyle f(x)=e^x[/spoiler]), then the derivative of that (and consequently the velocity versus time graph) will also be modeled by an exponential function. Consequently, the derivative of the velocity versus time graph will also be modeled by an exponential function. And so will the jerk.

What does this mean? What strange phenomena in nature is actually modeled like this?

Just back the fuck off. Every number up and down the number line thinks they can do what e does, but the minute things go exponential they run crying to the natural log function to make their bases line up.

OP, because that is the definition of e. Its purpose (among others) is to be the base of an exponential function so that the derivative of said function is equal to itself. It's a mathematical convenience. Just open up the fucking book.

>>3985104
>proof of e
>uses the natural logarithm to derive e.

UHHHHHH

Look OP, just google the shit out of leonhard euler and the natural number.

>>3985172

I am. But I'm also finding it interesting that most of you cannot explain this without clinging to your calculus and algebra.

>>3985188
>ask a question
>receive an answer
>y u use calculus and algebra? derp
dumb it down for me pl0x

>>3985158
this

>>3985188
>without clinging to your calculus and algebra

what the fuck are you talking about? what, you want some sort of special meaning to e that all the new age faggots tag onto the golden ratio?

constants in math are nothing special

>>3985211
Take a diff eq class once you pass high school, kid

>>3985203
>still can't explain the physical implications
>typical mathematician

e^x is the sum from n= 0 to infinity of x^n/n!
differentiating term by term we get
de^x/dx = sum from n = 0 to infinity of nx^(n-1)/n!
= sum from n = 1 to infinity of x^(n-1)/(n-1)!
= sum from n = 0 to infinity of x^n/n! = e^x

Tell you what: I'm just going to pose this question to a real physicist tomorrow and have him explain it.

He once told me, "You sometimes learn more about math in physics than you do in actual math classes."

How true...how true...

>>3985219

I'm glad that you can define e as a Taylor series. Really, I am.

But what I'm looking for is the physical ramifications.

>>3985233

f(t)=e^t

If f(t) is a function which describes the position of a particle at time t, then the velocity, acceleration, jerk, torque... etc, are all equal to the value of the position function.

>>3985104
(2) Didn't show or assume that the limit exists.
(3) Didn't show or assume that ln is continuous.
(5) Didn't show or assume that n is non-zero.

1. Take the Taylor series of e^(ix)
2. Take the Taylor series of cosx
3. Take the Taylor series of sinx. Multiply all the terms by "i".

Look for a pattern between the three.

>>3985229
Well that's why Ash is STILL trying to be a Pokemon Master. Gary, on the other hand, used the differential d/dy and fucked that exponential bitch up.

>>3985253

Proof was a loose term. I'm not interesting in playing semantic douchebaggery games that makes you feel superior because you think proofing is godlike. It's just the limit definition of e.

>>3985246

Thank you. You rock.

Are you able to define a physical situation where this actually applies (besides just the general motion of a particle)? Are there any interesting phenomena in nature that can be described like this?

>>3985233

Physical ramifications? What the fuck does that even mean?

Look, e is a nice number that comes up a lot naturally (THE NATURAL NUMBER HURRRRR) so its convenient to use when calculating many things. One example would be that the arcs of suspension bridges closely resemble the hyberbolic functions, which rely heavily on e.

What are the physical ramifications of pi? Its useful when working with circular shapes and happens to pop up randomly in a few places. Math constants aren't super heros kid.

>>3985253

Proof was a loose term. I'm not interested in playing semantic douchebaggery games that makes you feel superior because you think proofing is godlike. It's just the limit definition of e.

>>3985246

Thank you. You rock.

Are you able to define a physical situation where this actually applies (besides just the general motion of a particle)? Are there any interesting phenomena in nature that can be described like this?

>>3985246

>situation completely unfeasible
>OP is impressed

OK op. e can't be represented as a rational number, so if you can get some guy to bet you a million dollars that he can express it as a rational, you can make a million dollars!

>>3985271
magnets and/or gravity prly use it somehow

>>3985268

This is exactly what I wanted. e is useful when dealing with hyperbolic functions: especially when modeling cable suspension in electrical lines and bridge cables.

That is what I wanted to know: a physical application.

If I were asking about the physical ramifications of pi, circular applications (such as volume and area) would be what I would be looking for.

Of course, there are more complex applications where pi is used. I would be more interested in the complex applications just like I am more interested in more than just 'particle's motion'.

>>3985233
Why do you seem to think there are physical ramifications. All that would happen if you had a particle following an exponential path is the same as any other path, but its velocity would have the same curve as its position as well as acceleration etc etc. nothing particularly special.

>>3985271
Yes, Look up Bose-Einstein condensates. The mathematics might be a little heavy for you but basically particles in a BEC don't lose momentum to the outside environment due to the exponential nature of their motion (position, velocity, acceleration etc all described by e)

>>3985271

e isn't particularly interesting when it comes to direct physical properties of Nature. It is tangentially related to population growth and finances, as those models are exponential.

If you are interested in looking into relationships between mathematics and nature I recommend you turn your eye to phi.

<span class="math">\phi= \frac{1+\sqrt{5}}{2}[/spoiler]

Phi is the ratio of the (n+1)th to (n)th terms in the Fibonacci sequence. The sequence is produced by addition to previous terms, or a type of Growth. It is then only natural that we should see a ratio which reflects growth in a medium which is so fond of Growing :) (nature)

\frac{d}{dx}a^x=\lim_{h\to 0}\frac{a^{x+h}-a^x}{h}=\lim_{h\to 0}\frac{a^{x}a^{h}-a^x}{h}=a^x\left(\lim_{h\to 0}\frac{a^h-1}{h}\right).

The limit on the right-hand side is independent of the variable x: it depends only on the base a. When the base is e, this limit is equal to one, and so e is symbolically defined by the equation:

\frac{d}{dx}e^x = e^x.

>>3985295

Why else would it be convenient to have e as a mathematical constant if not to make physics easier to deal with.

Like this:

>>3985296

This is freaking awesome particularly because I'm starting on momentum in physics. The derivation of momentum from Newton's 2nd using time as bounds was quite interesting.

He briefly touched upon the derivation of energy and work using Newton's 2nd in a different way with position information.

So I can directly relate the lack of loss of momentum with the position function modeled by an exponential where it doesn't lose any of its velocity or acceleration.

Any further would probably be going way over my head. But to me it's just interesting to see applications like this to make math a little less abstract.

>>3985305
>phi
>important
>shows up in exactly ZERO physics equations, while e and pi are errywhere

face it : phi is the biology of mathematics. a derivative of the fundamental things that are bigger and better.

>>3985305
e is ubiquitous in physics
phi shows up occasionally in pentagons and petal arrangements
about 70% of what people claim about phi is bullshit

>>3985315

Here...

The limit on the right-hand side is independent of the variable x: it depends only on the base a. When the base is e, this limit is equal to one, and so e is symbolically defined by the equation:

<span class="math">\displaystyle \frac{d}{dx}a^x=\lim_{h\to 0}\frac{a^{x+h}-a^x}{h}=\lim_{h\to 0}\frac{a^{x}a^{h}-a^x}{h}=a^x\left(\lim_{h\to 0}\frac{a^h-1}{h}\right).[/spoiler]

<span class="math">\displaystyle \frac{d}{dx}e^x = e^x.[/spoiler]

>>3985305

This is mind boggling and so interesting. I remember briefly touching upon the Fibonacci sequence in precalculus during our discussion of the binomial coefficient and Pascal's triangle but that's pretty much it.

When will I be able to examine this behavior of phi? As in: what level of mathematics do you start dealing with phi? So far, this is the first time I've seen phi used and I'm in calculus III dealing with conversion from spherical coordinate systems to rectangular and cylindrical. That's pretty much the only time we've used phi.

How significant is e to physics? Just think it like that : e saves our as whenever there are some differential equations. And how significant are differential equations, both ordinary and partial, to physics?

>>3985329
>>3985332

Where did I mention importance? I was merely acknowledging the existence of Phi in nature.

e and pi do show, but they are more subtle and hidden. Things that are symmetric with or based on Phi stand out.

>>3985344
physics is 10000x more natural then what hippies and stoners call nature, aka random biology

>>3985339

Same poster here, I am actually in the EXACT same course you are in and the EXACT same subject material!

I have also not encountered Phi in my coursework. However, Next Semester I will be taking a course called Chaos theory. Some of the grad students I know hinted at possible Phi dabbling. I know for sure we are going to look at fractals.

>>3985342

So I'll have to wait till DEQ next semester to see some of the more interesting applications huh? Oh well. It's something to look forward to.

We just started taking partial derivatives and doing some work with differentials in calc. It's all very boring to me at the moment because I can't see many applications of it just yet.

We touched upon <span class="math">\displaystyle \nabla F(x,y,z)[/spoiler] (gradient) today. Just a few applications with it using planes and the tangential plane of intersection between different shapes.

>>3985356
Get used to thinking of that nabla as an operator if you haven't already.

>>3985369

Can you elaborate? Is that the same as thinking of it as the line that connects to planes?

>>3985233
Don't ask questions like
>Why is the derivative and integral of e itself?
If you're looking for physical ramifications. Physics involves a lot of maths, but maths involves absolutely no physics whatsoever.

>>3985339

The "phi" you use in the golden ratio is completely different from the "phi" you use in spherical coordinates in calc 3. The first one is a constant, the 2nd one is a variable of integration. You're confusing two completely unrelated concepts.

>>3985371

This is good advice. Noted for the future.

>2011
>not using the binomial expansion of (1+x/n)^n and double sums to show that the limit and series definitions of e^x are equivalent

small time

>>3985104
>using the natural logarithm to find the value of e
You get 1 point just for having the balls to post that on the internet.

Unfortunately, you lose all your points and go straight to hell for even implying that's a proof or derivation.

>>3985410
>really stretching the 2011 green text seriously hope meme
>2011

<div class="math">
(e^x)'
= \lim_{h\to0}\frac{e^{x+h}-e^x}h
= \lim_{h\to0}\frac{e^xe^h-e^x}h
= e^x\lim_{h\to0}\frac{e^h-1}h
</div><div class="math">
= e^x\lim_{h\to0}\frac1h\sum_{k=1}^\infty\frac{h^k}{k!}
= e^x\lim_{h\to0}\sum_{k=1}^\infty\frac{h^{k-1}}{k!}
= e^x\lim_{h\to0}\left(1+\sum_{k=2}^\infty\frac{h^{k-1}}{k!}\right)
</div><div class="math">
= e^x\Bigg(\underbrace{\lim_{h\to0}1}_{=1}+\underbrace{\lim_{h\to0}\sum_{k=2}^\infty\frac{h^{k-1}}{k!}
}_{=0}\Bigg)
</div><div class="math">=e^x</div>The integral version follows from the fundamental theorem of analysis,<div class="math">
\int_{-\infty}^x\mathrm d\xi\;e^\xi
= \int_{-\infty}^x\mathrm d\xi\;(e^\xi)'
= e^x</div>

>>3985605
>>3985605

isnt using the series expansion to prove a derivative a bit circular

>>3985632
Nope. You can define exp in terms of the series and not worry about Taylor. If you're then doing the Taylor expansion, it turns out that you recover your original series.
There may be other errors in there though, I just tex'd that in the 4chan window.

>>3985643

How do you find the series of a trascendent function without using Taylor? Im curious

>>3985655
By definition. You can define <span class="math">exp[/spoiler] in many ways, e.g.<span class="math">y'=y~;~y(0)=1[/spoiler], <span class="math">\exp(x)=\sum_k\frac{x^k}{k!}[/spoiler], <span class="math">y(a+b)=y(a)y(b)~;~y(0)=1[/spoiler], ...
... and one of them is the series representation.

>>3985605
What is it that you think you proved?
By the way, if you have the series expansion, you can simply differentiate it.

>>3985668 By the way, if you have the series expansion, you can simply differentiate it.
That would've saved me a few minutes, yes. Dammit

>>3985672
On the other hand, that proof works even if you don't know how to find the derivative of a series. Whether it's smart to replace exchanging differentiation/summation with limit/summation is a good question. Found a better proof for <span class="math">\lim\frac{e^z-1}z=0[/spoiler] in Amann-Escher, but it's in German.

I got some interesting answers today from the professor and he said exponentials show up quite a bit in differentials. He gave something cooling as an example (Newton's law of cooling) and he also mentioned something about a superconductor... Oh and someone in free fall where the horizontal asymptote happened to be the terminal velocity.

>>3985561

Found another assburger faggot lurking on /sci/.