Listen here, Little Dicky
Listen here, Little Dicky
Comments
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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
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is this how Brian Greene was born?
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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.
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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.
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yea, essentially, to me, calculus is like the study of slope and a slope of everything slope, with displacement, velocity, acceleration.
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Except you can kinda treat it as a fraction when dealing with differential equations
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Oh god this comment just gave me ptsd
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Only for separable equations
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And discrete math.
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It's not even a fraction, you can just cancel out the two "d"s
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"d"s nuts lmao
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Look it is so simple, it just acts on an uncountably infinite dimensional vector space of differentiable functions.
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It was a fraction in Leibniz’s original notation.
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And it denotes an operation that gives you that fraction in operational algebra...
Instead of making it clear that
dis an operator, not a value, and thus the entire thing becomes an operator, physicists keep claiming that there's no fraction involved. I guess they like confusing people.
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When a mathematician want to scare an physicist he only need to speak about ∞
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When a physicist want to impress a mathematician he explains how he tames infinities with renormalization.
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Only the sith deal in ∞
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....and Buzz Lightyear
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Why does using it as a fraction work just fine then?
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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
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Chicken thinking: "Someone please explain this guy how we solve the Schroëdinger equation"
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Is that Phill Swift from flex tape ?
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Little dicky? Dick Feynman?
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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.
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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.
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I've seen e^{d/dx}
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It legitimately IS exponentiation. Romanian lady was wrong.
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Let's face it: Calculus notation is a mess. We have three different ways to notate a derivative, and they all suck.
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clearly, d/dx simplifies to 1/x
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If not fraction, why fraction shaped?
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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.
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De dix, boss! De dix!
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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.
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Division is an operator
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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). -
Headache for mathematicians
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1/2 <-- not a number. Two numbers and an operator. But also a number.
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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.
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What is Phil Swift going to do with that chicken?
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The will repair it with flex seal of course
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Having studied physics myself I'm sure physicists know what a derivative looks like.
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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 fraction -
I still don't know how I made it through those math curses at uni.
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Calling them 'curses' is apt
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Software engineer: 🫦
Mathematicians will in one breath tell you in one breath they aren't fractions, then in the next tell you dz/dx = dz/dy * dy/dx
Brah, chain rule & function composition.
Have you seen a mathematician claim that? Because there's entire algebra they created just so it becomes a fraction.
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