8 ms·
Derivatives don't always act like fractions (2021)
- bsaul 2y agoi'm looking forward to the day calculus gets rewritten using more intuitive notations. Everytime i manipulate dx i feel like walking on a minefield.
- hyperbrainer 2y agoNewtonian notation certainly feels more elegant to me. But kind of painful to work with in LaTeX. Langrangian notation is almost the same, and much eaiser to type too.
- pseudostem 2y agoIt has been argued before [0] that Leibniz notation being embraced in mainland Europe and not adopted in England/UK was the reason England fell about a century behind. First heard of this in MIT Calc undergrad course on YouTube, but would be too tedious to find which video, hence ran a search on the Internet. [0] https://hsm.stackexchange.com/questions/7704/was-english-mathematics-behind-europe-by-many-years-because-of-newtons-notation https://hsm.stackexchange.com/questions/7704/was-english-mat...
- seanhunter 2y agoNewtonian notation is just doing time derivatives with a dot above them, so in Latex that is just \dot{x} = v . Which means dx/dt = v, or \ddot{x} = a. Did you mean "Leibniz's" notation[1]? If so, if you use the esdiff package[2] it's just \diffp{y}{x} for partials or \diff{x}{y} for regular derivatives. Lagrange's notation is when people do x' = v or x'' = a and Like the Newton's notation you kinda have to know from context that you are differentiating with respect to time unless they write it properly as a function with arguments which people often tend not to (at least I often tend not to I guess). Sometimes people call the partial derivative notation where you use subscripts "Lagrange's notation" also[3]. So like f_x(x,y) = blah is the partial derivative of f with respect to x. [1] Actually invented by Euler, or maybe some other guy called Arbogast or something[?sp] [2] https://ctan.math.illinois.edu/macros/latex/contrib/esdiff/esdiff.pdf https://ctan.math.illinois.edu/macros/latex/contrib/esdiff/e... [3] Even though that was also actually invented by Euler apparently.
- hyperbrainer 2y ago\dot {} is not convenient to write everytime, and I suck at remembering to use macros. On the other hand, just writeing f' is far faster.
- pif 2y ago> Everytime i manipulate dx i feel like walking on a minefield. Embrace the minefield, love the minefield! Signed a physicist
- xeonmc 2y agoSICM?
- Y_Y 2y agoIf you liked the "functional" style of calculus in SICM, or want a calculus only book in this vein I recommend Baby Spivak: https://en.wikipedia.org/wiki/Calculus_on_Manifolds_(book) https://en.wikipedia.org/wiki/Calculus_on_Manifolds_(book) (And obviously Functional Differential Geometry by the authors of SICM)
- leoc 2y agoCertainly Gerry Sussman's frustrations with ambiguous notation were a big reason for his decision to create SICM! https://youtu.be/arMH5GjBwUQ?t=236 https://youtu.be/arMH5GjBwUQ?t=236
- bsaul 2y agonever seen this talk before, thanks ! i feel less lonely.
- TachyonicBytes 2y agoYou can always try infinitesimal analysis[1] [1] https://people.math.wisc.edu/~hkeisler/calc.html https://people.math.wisc.edu/~hkeisler/calc.html
- Chris2048 2y agoI honestly don't know why infitesimals aren't widespread. It can basically have the same basis/justification can't it? But with the bonus of being more intuitive. You don't even need to use "infinity", it starts out as just a variable representing some unknown quantity, then you "round to zero" on output. I actually collected a bunch of old Infinitesimal calculus math books.
- Qem 2y ago> I honestly don't know why infitesimals aren't widespread. It can basically have the same basis/justification can't it? But with the bonus of being more intuitive. Indeed they are more intuitive, people like Newton and Leibniz invented/discovered calculus by thinking in terms of infinitesimals, but it took time to be made rigorous, in the XX century. By then network effects got we stuck with epsilons and deltas, given that was the approach made rigorous earlier, and broadly adopted, despite being more cumbersome.
- TachyonicBytes 2y agoWould you mind giving us the titles of those books?
- Chris2048 2y agoThey are in the attic at the moment, but they are all fairly old books (and terse, dry, basic formatting/illustration), seemingly from a period in time when infitesimals were apparently more popular. There are a few similar ones on IA, e.g. https://archive.org/details/in.ernet.dli.2015.148501/page/n81/mode/1up https://archive.org/details/in.ernet.dli.2015.148501/page/n8... On that page the 'h' term is the infitesimal, as in d(x^2) / dx = 2x + h Though I prefer something like 'Δx' to make the link to x more explicit. Would love to see a more modern book on the topic.
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- BobaFloutist 2y agoThe XKCD about unifying standards under a new standard is how I feel every single time I learn anew piece of math notation. "This is ridiculous! We need a better, more intuitive notation that's also easier to do math at."
- fragmede 2y agoThat's xkcd 927,a mnemonic for that is 3^2 * 100 + 3^3
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- BobbyTables2 2y agoI’d be thrilled if mathematicians would just use multicharacter variable names instead of getting overly fancy with diacritics and italic/bold/capital/Greek variations.
- semi-extrinsic 2y agoThe key clarification is in one of the comments: if you want to treat partial derivatives like fractions, you need to carry the "constant with respect to foo" modifier along with both nominator and denominator. Once you do that, it's clear that you can't cancel "dx at constant z" with "dx at constant y" etc. And then the remaining logic works out nicely (see thermodynamics for a perfect application of this).
- slooonz 2y agoI still don’t understand what "at constant something" means. I mean formally, mathematically, in a way where I don’t have to kinda guess what the result may be and rely on my poor intuitions and shoot myself continually in the foot in the process. Does someone has a good explanation ?
- siev 2y agoImagine a function z=f(x,y) in 3D space. Now picture a plane at say, x=3, that is parallel to the plane passing through the Y and Z axes. This x=3 plane cuts through our function, and its intersection with the z=f(x,y) function forms a sort of 2D function z=g(x)=f(3,y). (The Wikipedia page[1] has nice images of this [2]) The slope of this new 2D function on the x=3 plane at some point y is then the partial derivative ∂z/∂y for constant x at the point (3,y). As we are "fixing" the value of x to a constant, by only considering the intersection of our original function with a plane at x=x_0. [1] https://en.wikipedia.org/wiki/Partial_derivative https://en.wikipedia.org/wiki/Partial_derivative [2] https://en.wikipedia.org/wiki/File:Partial_func_eg.svg https://en.wikipedia.org/wiki/File:Partial_func_eg.svg
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- otisv 2y agoI think you mean z=g(y)=f(3,y) instead of g(x)
- slooonz 2y agoThat’s just the standard partial derivative in multivariable calculus. This one I have no trouble to understand. My question is about "at constant something" as used in thermodynamics, where "at constant something" is clearly doing more work than just "partial derivative". What work ? How ? Damned if I know. Consider f(x,y,z), let’s say f(x, y, z) = x^2 + 3y^3 - e^(-z). What’s the difference between "the partial derivative of f with respect to x" and "the partial derivative of f with respect to x at constant y" ? The first one is already at constant y ! In standard multivariate calculus, the partial derivative of f with respect to x , as you explained, is always "at constant y and z". In thermodynamics, you can say things like "partial derivative of pressure with respect to volume" and add "at constant temperature" or "at constant entropy" and get different results. What ? Why ? How ?
- rob_c 2y agoAm I missing something, I don't see how the examples are more "intuitive" as they just provide an allied example of using this? My pain was always Hamiltonians and Legendre equations for systems because the lecturer believed in learn by rote rather than explaining something that I'm sure for him was simply intuitive.
- ttoinou 2y agoWhy would you even tell in the first places derivatives are simply fractions ? They’re not, unless in some very specific physical approximations and in that case don’t try to do anything funky, sticks with the basics stuff
- ajkjk 2y agoWell the fact that they're often written as fractions might be one reason...
- seanhunter 2y agoMy understanding is they actually are fractions of things called differential one-forms[1], but even most people who can do calculus don't get to differential geometry, so the sense in which they are fractions is not commonly understood. Michael Penn explains it here https://youtu.be/oaAnkzOaNwM?si=nwNNg4pl7WW4KvIO https://youtu.be/oaAnkzOaNwM?si=nwNNg4pl7WW4KvIO [1] https://mathworld.wolfram.com/Differentialk-Form.html https://mathworld.wolfram.com/Differentialk-Form.html
- jasomill 2y agoA 1-form is a section[1] of the cotangent bundle[2] of a manifold. In other words, a rank 1 covariant tensor field. At any given point p on an n-dimensional manifold, a 1-form defines an n-dimensional cotangent vector (in the language of bundles[3], a point in the fiber over p). So how do we define fractions of sections or vectors? In the article, Baez defines fractions of 2-forms on the plane as the pointwise ratio of coefficients of a basis vector, which he can do because, as he points out, the space of 2-forms at a point on a 2-dimensional manifold is a 1-dimensional vector space (more generally, for k-forms on an n-dimensional manifold, this dimension is n choose k, so only 1 for 0-forms [functions] and n-forms). [1] https://mathworld.wolfram.com/BundleSection.html https://mathworld.wolfram.com/BundleSection.html [2] https://mathworld.wolfram.com/CotangentBundle.html https://mathworld.wolfram.com/CotangentBundle.html [3] https://mathworld.wolfram.com/FiberBundle.html https://mathworld.wolfram.com/FiberBundle.html
- brooke2k 2y agothey refer in the beginning to physics classes, and I had the same exact experience in university. diffeq was not a prereq and yet instead of explaining the derivation of these equations, our physics professor just handwaved and said "they're basically just fractions, don't think about it too much"
- xorvoid 2y agoI’ve never liked to conflation with fractions. Abuse of notation. And it causes so much confusion. Also integrals with “integrate f(x) dx” where people treat “dx” as some number than can be manipulated, when it’s more just part of the notation “integrate_over_x f(x)” Sigh. These are sadly some kind of right-of-passage, or mathematical hazing. Sad.
- setopt 2y agoI consider it not an abuse of notation but a helpful notation. It insinuates correctly >90% of calculus rules, which tend to be hard to remember otherwise.
- nyeah 2y agoOrdinary derivatives work fine as fractions. They are rigorously the limit of a fraction. Same deal with dx inside the integral, it is rigorously the limit of a small \Delta x in a summation. Baez is mixing partial derivatives with different variables treated as constants. Whole different ball game.