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But df/dx is a fraction, is a ratio between differential of f and standard differential of x. They both live in the tangent space TR, which is isomorphic to R.
What’s not fraction is \partial f / \partial x, but likely you already know that. This is akin to how you cannot divide two vectors.
The thing is that it’s legit a fraction and d/dx actually explains what’s going on under the hood. People interact with it as an operator because it’s mostly looking up common derivatives and using the properties.
Take for example
∫f(x) dxto mean "the sum (∫) of supersmall sections of x (dx) multiplied by the value of x at that point ( f(x) ). This is why there’s dx at the end of all integrals.The same way you can say that the slope at x is tiny f(x) divided by tiny x or
d*f(x) / dxor more traditionally(d/dx) * f(x).The other thing is that it’s legit not a fraction.
it’s legit a fraction, just the numerator and denominator aren’t numbers.
No 👍
try this on – Yes 👎
It’s a fraction of two infinitesimals. Infinitesimals aren’t numbers, however, they have their own algebra and can be manipulated algebraically. It so happens that a fraction of two infinitesimals behaves as a derivative.
Ok, but no. Infinitesimal-based foundations for calculus aren’t standard and if you try to make this work with differential forms you’ll get a convoluted mess that is far less elegant than the actual definitions. It’s just not founded on actual math. It’s hard for me to argue this with you because it comes down to simply not knowing the definition of a basic concept or having the necessary context to understand why that definition is used instead of others…
Why would you assume I don’t have the context? I have a degree in math. I could be wrong about this, I’m open-minded. By all means, please explain how infinitesimals don’t have a consistent algebra.
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I also have a masters in math and completed all coursework for a PhD. Infinitesimals never came up because they’re not part of standard foundations for analysis. I’d be shocked if they were addressed in any formal capacity in your curriculum, because why would they be? It can be useful to think in terms of infinitesimals for intuition but you should know the difference between intuition and formalism.
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I didn’t say “infinitesimals don’t have a consistent algebra.” I’m familiar with NSA and other systems admitting infinitesimal-like objects. I said they’re not standard. They aren’t.
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If you want to use differential forms to define 1D calculus, rather than a NSA/infinitesimal approach, you’ll eventually realize some of your definitions are circular, since differential forms themselves are defined with an implicit understanding of basic calculus. You can get around this circular dependence but only by introducing new definitions that are ultimately less elegant than the standard limit-based ones.
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Headache for mathematicians
The world has finite precision. dx isn’t a limit towards zero, it is a limit towards the smallest numerical non-zero. For physics, that’s Planck, for engineers it’s the least significant bit/figure. All of calculus can be generalized to arbitrary precision, and it’s called discrete math. So not even mathematicians agree on this topic.
1/2 <-- not a number. Two numbers and an operator. But also a number.
In Comp-Sci, operators mean stuff like
>,*,/,+and so on. But in math, an operator is a (possibly symbollic) function, such as a derivative or matrix.Youre not wrong, distinctively, but even in mathematics “/” is considered an operator.
https://en.m.wikipedia.org/wiki/Operation_(mathematics)
oh huh, neat. Always though of those as “operations.”
We teach kids the derive operator being
'or·. Then we switch to that writing which makes sense when you can use it properly enough it behaves like a fractionSoftware engineer: 🫦
clearly, d/dx simplifies to 1/x
If not fraction, why fraction shaped?
Mathematicians will in one breath tell you they aren’t fractions, then in the next tell you dz/dx = dz/dy * dy/dx
Not very good mathematicians if they tell you they aren’t fractions.
Have you seen a mathematician claim that? Because there’s entire algebra they created just so it becomes a fraction.
Brah, chain rule & function composition.
Also multiplying by dx in diffeqs
vietnam flashbacks meme
This is until you do multivariate functions. Then you get for f(x(t), y(t)) this: df/dt = df/dx * dx/dt + df/dy * dy/dt
(d/dx)(x) = 1 = dx/dx
Having studied physics myself I’m sure physicists know what a derivative looks like.
This very nice Romanian lady that taught me complex plane calculus made sure to emphasize that e^j*theta was just a notation.
Then proceeded to just use it as if it was actually eulers number to the j arg. And I still don’t understand why and under what cases I can’t just assume it’s the actual thing.
e𝘪θ is not just notation. You can graph the entire function ex+𝘪θ across the whole complex domain and find that it matches up smoothly with both the version restricted to the real axis (ex) and the imaginary axis (e𝘪θ). The complete version is:
ex+𝘪θ := ex(cos(θ) + 𝘪sin(θ))
Various proofs of this can be found on wikipeda. Since these proofs just use basic calculus, this means we didn’t need to invent any new notation along the way.
I’m aware of that identity. There’s a good chance I misunderstood what she said about it being just a notation.
It’s not simply notation, since you can prove the identity from base principles. An alien species would be able to discover this independently.
It legitimately IS exponentiation. Romanian lady was wrong.
It is just a definition, but it’s the only definition of the complex exponential function which is well behaved and is equal to the real variable function on the real line.
Also, every identity about analytical functions on the real line also holds for the respective complex function (excluding things that require ordering). They should have probably explained it.
She did. She spent a whole class on about the fundamental theorem of algebra I believe? I was distracted though.
Let’s face it: Calculus notation is a mess. We have three different ways to notate a derivative, and they all suck.
Calculus was the only class I failed in college. It was one of those massive 200 student classes. The teacher had a thick accent and hand writing that was difficult to read. Also, I remember her using phrases like “iff” that at the time I thought was her misspelling something only to later realize it was short hand for “if and only if”, so I can’t imagine how many other things just blew over my head.
I retook it in a much smaller class and had a much better time.
I’ve seen e^{d/dx}
Why does using it as a fraction work just fine then? Checkmate, Maths!
It doesn’t. Only sometimes it does, because it can be seen as an operator involving a limit of a fraction and sometimes you can commute the limit when the expression is sufficiently regular
Added clarifying sentence I speak from a physicists point of view.
I found math in physics to have this really fun duality of “these are rigorous rules that must be followed” and “if we make a set of edge case assumptions, we can fit the square peg in the round hole”
Also I will always treat the derivative operator as a fraction
2+2 = 5
…for sufficiently large values of 2
i was in a math class once where a physics major treated a particular variable as one because at csmic scale the value of the variable basically doesn’t matter. the math professor both was and wasn’t amused
Engineer. 2+2=5+/-1
I mean as an engineer, this should actually be 2+2=4 +/-1.
Computer science: 2+2=4 (for integers at least; try this with floating point numbers at your own peril, you absolute fool)
Freshmen engineer: wow floating point numbers are great.
Senior engineer: actually the distribution of floating point errors is mindfuck.
Professional engineer: the mean error for all pairwaise 64 bit floating point operations is smaller than the Planck constant.
0.1 + 0.2 = 0.30000000000000004
comparing floats for exact equality should be illegal, IMO
Statistician: 1+1=sqrt(2)
pi*pi = g
units don’t match, though
Found the engineer
I always chafed at that.
“Here are these rigid rules you must use and follow.”
“How did we get these rules?”
“By ignoring others.”
is this how Brian Greene was born?
Derivatives started making more sense to me after I started learning their practical applications in physics class.
d/dxwas too abstract when learning it in precalc, but once physics introducedd/dt(change with respect to time t), it made derivative formulas feel more intuitive, like “velocity is the change in position with respect to time, which the derivative of position” and “acceleration is the change in velocity with respect to time, which is the derivative of velocity”Possibly you just had to hear it more than once.
I learned it the other way around since my physics teacher was speedrunning the math sections to get to the fun physics stuff and I really got it after hearing it the second time in math class.
But yeah: it often helps to have practical examples and it doesn’t get any more applicable to real life than d/dt.
I always needed practical examples, which is why it was helpful to learn physics alongside calculus my senior year in high school. Knowing where the physics equations came from was easier than just blindly memorizing the formulas.
The specific example of things clicking for me was understanding where the “1/2” came from in distance = 1/2 (acceleration)(time)^2 (the simpler case of initial velocity being 0).
And then later on, complex numbers didn’t make any sense to me until phase angles in AC circuits showed me a practical application, and vector calculus didn’t make sense to me until I had to actually work out practical applications of Maxwell’s equations.
yea, essentially, to me, calculus is like the study of slope and a slope of everything slope, with displacement, velocity, acceleration.
