/sci/ - Science & Math

[Anonymous No.11129152]

If a derivative is the slope of an equation, then a differential is the...?

[Anonymous No.11129157]

>>11129152
...same as the derivative. What are you asking?

[Anonymous No.11129162]

[math]y=A+B x_1 +C x_2[/math]

[Anonymous No.11129163]

>>11129162
Oops no A

[Anonymous No.11129295]

Not sure what the fuck are you asking either, but I believe you should read either of those links
https://en.wikipedia.org/wiki/Partial_derivative
https://en.wikipedia.org/wiki/Differential_form

[Anonymous No.11129437]

>>11129152
it's a mathematical operator

[Anonymous No.11129506]

>>11129152
In introductory calculus they tell you not to consider dy/dx a fraction but for practical purposes you often do need to treat it as a fraction and if you intuitively consder dy/dx a ratio of two infinitesimal quanities, you will not be wrong. Cue the math nerds who are going to swarm in and give me shit.

[Anonymous No.11129521]

I forgot to explicitly say what a differential is. A differential is just an infinitesimal interval [math]\Delta x[/math]. That is all it means. Just a very tiny change in x. If you have ever done problems involving estimating using differentials in Calc 3 you will notice that it usually just involves using the linearization (tangent plane) of a function to estimate the value of a function near a point. The reason it works as a good estimate of the function is because we take [math]\Delta x[/math] to be very tiny.

[Anonymous No.11129646]

it´s the same

[Anonymous No.11129671]

>>11129506
>infinitesimal

Stop right there criminal scum, that's a forbidden word. Cauchy didn't formalize calculus for nothing you know.

[Anonymous No.11129843]

>>11129671
newton would have called Cauchy a brainlet and then waited for him to die before publishing a book on why he's wrong

[Anonymous No.11129879]

>>11129152
A differential is like a 1-tensor, it wants to "eat" a vector and return a number, but if by differential you mean the exterior differentiation of a 0-form, which is just a plain function, then it could be thought just as a the gradient of the function, which simply points in the direction where the functions grows.

[Anonymous No.11130109]

>>11129152
It's the best linear approximation near a point. Fix a point [math](x,y)[/math] and consider an "increment" [math](x+h,y+k)[/math]. We can develop [math]f(x+h,y+k)[/math] into a Taylor series:
[eqn]f(x+h,y+k) = f(x,y) + \text{linear term in }(h,k) + \text{quadratic term in }(h,k) + \dots[/eqn]
differential is just the linear term:
[eqn]f(x+h,y+k) = f(x,y) + df_{(x,y)}(h,k) + \text{hot}(h,k) [/eqn]
with the explicit formula given by
[eqn]df_{(x,y)}(h,k) = \frac{\partial f}{\partial x}(x,y)\cdot h+\frac{\partial f}{\partial y}(x,y)\cdot k[/eqn]

[Anonymous No.11130120]

>>11129152
I thought they were the same thing.

[Anonymous No.11130130]

>>11130120
strictly speaking, derivative is a number, differential is a linear map. however any linear map R -> R is just a multiplication by a single number. you can guess what this number is.

[Anonymous No.11130139]

>>11130130
When is this distinction important?

[Anonymous No.11130153]

>>11130139
when you move to more dimensions. the derivative for a single-variable function is:
1/ limit of the difference quotient f(x+h) - f(x) / h
2/ slope of the tangent line
none of this has a straightforward generalization to more dimensions. however the differential, defined to be the best linear approximation, generalizes without problems.

[Anonymous No.11130161]

>>11130153
>none of this has a straightforward generalization to more dimensions.
I thought that's exactly what a tangential plane does. The differential generalizing to the vector describing slope of the tangential plane.

[Anonymous No.11130168]

>2019
>still not teaching the geometrical intuition behind derivatives in schools
STEM is kill.

[Anonymous No.11130170]

>>11130168
But they do?

[Anonymous No.11130176]

>>11130161
you have a tangent line to the graph of f : R -> R. you can describe it by its slope - tan of the angle between the line and the x-axis. this is the derivative. for a function of two variables, you have the tangent plane. however you don't know what "a slope" is here. it's not a good idea to define the plane by some angle or something.

let's start again. you have a tangent line to the graph of f : R -> R. this line can be described as a graph of a linear map which is a very good approximation of f. this linear map is the differential. for a function of two variables you have the tangent plane. this tangent plane can just as well be described as a graph of a linear map which is a good approximation to this function. this is how the differential generalizes immediately.

[Anonymous No.11130196]

>>11130161
>>11130176
Also I think that differential starts to make better sense when one considers mappings [math]\mathbb{R}^m \to \mathbb{R}^n[/math], not just into [math]\mathbb{R}[/math] . For example [math]\mathbb{R^2} \to \mathbb{R^2}[/math] is some, non-linear in general, transformation of the plane. Here the derivative, as a slope of something, doesn't make sense all. It's also not a good idea to try to understand this map via its graph which now lives in [math]\mathbb{R^2} \times \mathbb{R^2} = \mathbb{R^4}[/math]. The differential, as the best linear approximation, makes perfect sense:

Look at pic related. Blue grid is the original non-linear transformation (before and after), i.e. a map [math]\mathbb{R^2} \to \mathbb{R^2}[/math]. Red grid is a linear transformation which approximates the blue one near the green point. This is the differential.

[Anonymous No.11130231]

Differential is a property of the function itself. For example, a function such as ln(x) has no derivative at x = 0; Basically, t means that if you know a function, then under certain conditions you should know its n-th derivative(excluding functions that can be represented as Taylor Polynomials). By the Fundamental Theorem of Calculus, this means that if you know the n-th derivative, then you know a bit about the integral of that function. Basically this means that functions with certain properties are categories on their own. As it turns out, there are more functions that have differentials than otherwise. Even in very chaotic functions, there are intervals where the function will have differentiable curves. meaning that chaotic functions are really just a combination of differentiable curves and irregular/ nondifferentiable curves. Because you have a combination of differentiable curves, these small intervals contribute a larger cardinality of points enclosed in a smooth curves. There are two cases to look at. First is smooth curves with no known function. Well it shouldn't be difficult to see that some portions can be approximated via curve fitting. Here, the vast majority of points are enclosed in a curve.

[Anonymous No.11130422]

>>11129521
There's way more to it than that tbqh. It's sufficient for someone just learning the concepts though.

[Anonymous No.11130473]

>>11130153
Somebody doesn't know what the tangent bundle is.... You can define it for any general manifold.

[Anonymous No.11130485]

>>11130422
Care to elaborate? When is [math]\Delta x[/math] not sufficient?

[Anonymous No.11130495]

>>11130473
and your point is ?

[Eris Omniquery No.11130530]

>>11130168
>2019
>Still not teaching the physical intuition behind calculus, with the operations of integration and differentiation corresponding to two fundamental reference-frames of change-perception roughly corresponding to "instantaneous change in the present" and "cumulative change over time."
>Not realizing that this applies to epistemology, with integration corresponding to "understanding" and differentiation to "discovery."
>Not realizing that this is the foundation of a theory of experiential evolution and a way to turn psychology into a mathematical physics.
Daily reminder that calculus is "the study of change."

[Anonymous No.11130568]

>>11130170
Errr, no. Not at all.

[Anonymous No.11130661]

>>11129152
Base vector of the tangent space.

[Anonymous No.11130670]

>>11130495
He didn't have one, he just wanted to be a smug cunt.

[Anonymous No.11130675]

>>11129152
How the fuck does an equation have a slope?

What this guys said:>>11130109

[Anonymous No.11130706]

Don't treat derivatives as fractions or 'small change in x,y'. They are operators.

Simple multiplicative properties will show why you can't treat derivatives as small changes in x,y.

[Anonymous No.11130723]

>>11130670
he succeeded in making a fool of himself then

[Anonymous No.11130734]

>>11130485
Try doing something similar for second order or higher. It only works as a fraction for first order derivatives because they're linear.

[Anonymous No.11130806]

>>11130706
What is an operator?

[Anonymous No.11130927]

For an affine variety X, the differential is an isomorphism between the tangent space at a point x in X and the vector space of linear forms on m/m^2, where m is the maximal ideal of the local ring at x.

[Anonymous No.11131071]

>>11130927
nobody cares, anon-kun

[Anonymous No.11131189]

>>11131071

t. engineer

[Anonymous No.11131268]

>>11130161
>... tangential plane does. The differential generalizing to the vector describing slope of the tangential plane.

I think the generalization of the idea behind the derivative from 1D calculus, which I believe can be interpreted as "how much this function grows", is the covariant derivative in general for any manifold, for R^n its just the directional derivative, how much a vector field X changes along this other vector field Y, defining a tangent plane at each point will only define the tangent bundle, which does not tell you the whole story about how much the function changes.

[Anonymous No.11131286]

>>11131268
Just in case anybody missed it, this whole post is one single sentence.
I agree with the content though.

[Anonymous No.11131455]

>infinitesimal

STOP USING THAT WORD YOU PRETENTIOUS CUNT

No one who is taking Calc 1 uses non-standard analysis.

[Anonymous No.11132192]

>>11130806
a function that maps another function from one set of functions to a function in another set of functions (or, often, the same set)

[Anonymous No.11133007]

>>11129157
A derivative is a ratio of differentials. How are they the same?

[Anonymous No.11133041]

>>11129671
>Cauchy didn't formalize calculus for nothing you know.
I imagine his motivation was taking an intuitively clear and effective way of thinking about things and shitting all over it.
Infinitesimals make perfect sense. You can't divide by 0 so calculus is born from doing what you need to solve those equations by getting an arbitrarily small nonzero stand-in value over to a safe place where it's not a divisor so you can eliminate it and see the answer.
Limits meanwhile are obnoxious satanic bullshit meant to confuse and irritate you. I have no idea how anyone is able to comfortably think in terms of that fucked up needlessly convoluted reworking of what used to be a pure and sensible infinitesimal based system.

[Anonymous No.11133082]

>>11131455
Infinitesimal is a quantity that cannot be split any further. 1/infinity. I at least know that from the Leibnizian formulation of Calculus, despite not having taken anything but Calculus I so far
Having said that, I don't think it's possible to deal with infinitesimally small quantities except on a merely hypothetical level.

[Anonymous No.11133090]

>>11130530
They do teach it that way. That's why high school students and college freshmen usually take Physics alongside Calculus, for the sake of highlighting the importance of derivatives in figuring out the degree of change in some ongoing process, and integration in adding up quantities of change.
I'm no mathematician, but I at least know that.

[Anonymous No.11133105]

>>11129152
You are not expected to understand, but to perform the operations you memorized to pass the examination, to show your conformity to the norm.

[Anonymous No.11133187]

>>11130530
Geometrical, not physical.
Physical follows from the geometrical intuition. Doing it the other way is confusing when changing to functions or multiple variables.