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Headache for mathematicians
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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Software engineer: 🫦
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 fractionIf 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.
Having studied physics myself I’m sure physicists know what a derivative looks like.
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
I always chafed at that.
“Here are these rigid rules you must use and follow.”
“How did we get these rules?”
“By ignoring others.”
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.
Look it is so simple, it just acts on an uncountably infinite dimensional vector space of differentiable functions.
When a mathematician want to scare an physicist he only need to speak about ∞
When a physicist want to impress a mathematician he explains how he tames infinities with renormalization.
Is that Phill Swift from flex tape ?
What is Phil Swift going to do with that chicken?
Chicken thinking: “Someone please explain this guy how we solve the Schroëdinger equation”
Little dicky? Dick Feynman?
