/sci/ - Science & Math

dx might look like two operators (d and x), but it is actually one, 'dx'. dx stands for the infinitesimal of x.

>Is it just notation?

Yes, and it's a useful notation. You will learn to appreciate it later.

>>5654551
But dy is the infinitesmal of y? So wouldn't the same principle follow?

First year calculus here, just asking.

>>5654551
But why is it not [/math]dy^2/dx^2[/math] ?

d/dx ( d/dx (f(x))) = d2/(dx)2 f(x)

>>5654557
http://en.wikipedia.org/wiki/Leibniz%27s_notation

>>5654536
if you think about it unit-wise it makes sense. I.E
displacement = x [=] L
velocity= dx/dt [=] L/T
acceleration= d^2x/dt^2 [=] L/T^2

>>5654556
Because y in a function in this context.

>>5654557
It can be, depends on the context. When you learn about seperable differential equations you will learn to use dy in this context as a single operator.

Think of y = f(x) and there being d/dx * d/dx * f(x)

>>5654536
I was confused about this for the longest time (seven years?) before I finally realized how dumb the notation is. The idea is that <div class="math">
\frac{\mathrm{d}}{\mathrm{d} x},
</div> represents a differentiation operator, which sends the function <span class="math">y[/spoiler] to its derivative <span class="math">\frac{\mathrm{d}}{\mathrm{d} x} y = y'.[/spoiler] The second derivative operator is then simply <div class="math">
\left( \frac{\mathrm{d}}{\mathrm{d} x} \right)^2,</div> where the superscript two does not indicate squaring the operator, but rather applying the operator twice. Because users of Leibniz notation want to view the numerator and denominator as separate, they would write <div class="math">\left(\frac{\mathrm{d}}{\mathrm{d} x}\right)^2 = \frac{\mathrm{d}^2}{(\mathrm{d} x)^2}.</div>
Because users of Leibniz notation are lazy (not a bad thing at all), they throw away the parentheses. This is a good thing, because the notation doesn't make too much formal sense in the first place, so it's better to just make it simpler to write. We arrive at <div class="math"> \left(\frac{\mathrm{d}}{\mathrm{d} x}\right)^2 = \frac{\mathrm{d}^2}{(\mathrm{d} x)^2} = \frac{\mathrm{d}^2}{\mathrm{d} x^2}.
</div>
When the second derivative operator is applied to the function <span class="math">y[/spoiler], one finally gets <div class="math">
y'' = \frac{\mathrm{d}^2}{\mathrm{d} x^2} y = \frac{\mathrm{d}^2 y}{\mathrm{d} x^2}.
</div>

>>5654635
I can't believe I actually learned something on 4chan.

>>5654635
bravo sir

>>5654635
Good explanation.

It's actually incredible how poorly calculus is taught. Nearly every student struggles with this very simple concept.

>>5654635
>taking the time to put the letter d in \mathrm even when TeXing in mathjax

son, i am proud

>>5654656
When you finally see an expression like
<span class="math">\displaystyle \frac{\partial ^2 f}{partial x^2}[/spoiler]
vs. something like
<span class="math">\displaystyle \frac{\partial ^2 f}{partial x \partial y}[/spoiler]
I think the idea of differential operator starts to make more sense as compared to the single-variable point of view where (for me personally) it was just like 'fuck this liebniz shit why don't i stick primes and double primes everywhere and what do you mean it's incorrect to say the derivative of x^2 or that it ISN'T just a formality to put dx after my integrals'

>>5654677
fuck me I always mess up something or other
<span class="math">\displaystyle \frac{\partial ^2 f}{\partial x^2}[/spoiler]
<span class="math">\displaystyle \frac{\partial ^2 f}{\partial x \partial y}[/spoiler]

>>5654649
It does happen on occasion on /sci/ brah.

>>5654677
>>5654681

Its painful to watch you struggle, man

>>5654635
Already knew that, but that won't stop me thanking you anon. Good work.

>>5654557
>First year calculus here

>graduate student pretending to be a undergraduate.

nothing new.

>>5654635
this notation have/had a real meaning.

d is for a difference

df/dx = [f(x+dx)-f(x)]/dx and since you say that dx is an Infinitesimal it is logical to say that df/dx=f' ( because f'(x) is the limit ( if it exist and is finit ) of [f(x+h)-f(x)]/h when h approaches 0, by definition )

if you applie d to df then you found d(df)=d2f=[f(x+2dx)-f(x+dx)]-[f(x+dx)-f(x)]=f(x+2dx)-2f(x+dx)+f(x)
once again you can prove that if f is C2 then f''(x) is the limit of [f(x+2h)-2f(x+h)+f(x)]/h2 when h approaches 0 and thus the notation d2f/dx2 for f'' is justified.

anyway df2/dx2 would be wrong because it mean [f(x+dx)-f(x)]2/dx2 which isn't, in general, equal to f'' :
[f(x+h)-f(x)]2=f(x+h)2-2f(x+h)f(x)+f(x)2 which is different from f(x+2h)-2f(x+h)+f(x)

help

>>5654635
> where the superscript two does not indicate squaring the operator, but rather applying the operator twice.
What would squaring a unary operator mean other than applying it twice, I'd like to know.

>>5656366
x^2 = x*x

You can't square operators.

>>5656390
It's called composition, champ.

>>5656402
What are you failing to understand here? Squaring is an operation. You can't operate upon operations.

>>5656425
you can square the result

>>5656425
<span class="math">\mathrm{f}^{n}(x) = \mathrm{f}(\mathrm{f}^{n-1}(x))[/spoiler]
deal with it

>>5656366
>What would squaring a unary operator mean other than applying it twice, I'd like to know.
The point is that the superscript two indicates function composition. And the "does not indicate squaring" may be poorly worded, but is not unnecessary. Consider, for example, the following.
Since <span class="math">\mathbb{R}[/spoiler] is a vector field, any function <span class="math">f:\mathbb{R}\to \mathbb{R}[/spoiler] is in principle an operator.
The function <span class="math">\sin:x\mapsto \sin \, x[/spoiler] is therefore an example of an operator from <span class="math">\mathbb{R}[/spoiler] to itself. But what does the notation <span class="math">\sin^2[/spoiler] indicate? Should we interpret <span class="math">\sin^2x[/spoiler] as <div class="math">
\sin^2 x = \sin(\sin(x)),
</div> or as <div class="math">
\sin^2 x = \left( \sin \, x \right)^2,
</div> for example? While the former might make more sense, the latter is the most popular convention.
The significance of this is that "squaring" of operators sometimes does not denote function composition, but other things, which may also be considered some form of "squaring". Since this sort of notation is prevalent, it's not unnecessary to be rather explicit.

>>5656478
No, my question was meant to be, "Faced with this notation, what else could it mean?"

> any function which maps a real to a real is in principle an operator
Indeed, which is why I dislike the standard interpretation of <span class="math">\sin^{n}(x)[/spoiler]. For people who like generalization, mathematicians are remarkably irritating about special cases of notation.

>>5654649
/sci/ is very capable of being helpful when it wants to be...

... and when its not getting spammed with homework problems and troll science

>>5656486
Do you even read? That entire argument is applicable to the derivative. The notation could mean taking the derivative of the derivative, or squaring the derivative.

>>5656493
You said "does not indicate squaring the operator." What does it mean, ever, to square an operator, but function composition?

Can you read?

>>5656498
He didn't say "does not indicate squaring the operator". I did, but it doesn't matter.
I pointed out above that in the case of the sine function, a superscript two can have different interpretations. When talking about the square of an operator, I agree that one normally refers to function composition. It can (and perhaps should) be argued that the square of the operator <span class="math">x\mapsto \sin(x),[/spoiler] is the operator <span class="math">x\mapsto \sin(\sin(x)).[/spoiler] Furthermore, it is arguable that the operator <span class="math">x\mapsto \left( \sin(x) \right)^2[/spoiler] isn't really a square of the sine operator at all. But the confusing notation exists, and if you ask a random person for the square of the sine function, you'll most likely get the function <span class="math">x\mapsto \left( \sin(x) \right)^2.[/spoiler] Because of this, it wouldn't be implausible (but perhaps naïve) for someone to believe that <span class="math">\left( \frac{\mathrm{d}}{\mathrm{d} x} \right)^2 [/spoiler] is to be interpreted as <div class="math">
\left( \frac{\mathrm{d}}{\mathrm{d} x} \right)^2 : f \mapsto \left( \frac{\mathrm{d}}{\mathrm{d} x}
f \right)^2, </div> or perhaps something even stranger. Other interpretations (than composition) for the "square" of the differential operator obviously exist, and therefore I tried to remark that the superscript two should not be thought of as a "square" of the operator, but as function composition, and nothing else. Thinking of "superscript two" as "square" might be notationally and conceptually neat, but adds an unnecessary layer of complication to someone who wants Leibniz's notation explained to him or her.

dy/dx notation is the worst bullshit ever, stop trying to understand it - it's all ad-hoc crap

>>5656425
I can. Someday, maybe you can too (see - all of modern mathematics).

>>5656627

Makes perfect sense to me. What's a more intuitive way to describe rate of change of one quantity with respect to another?

>>5656627
erhm, i disagree. useful shorthand for THIS use, no?

Mathematical rigor is not needed in every bit of notation as long as you know what you are doing and what you can/cant do!