[HN Gopher] Derivatives don't always act like fractions (2021)
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Derivatives don't always act like fractions (2021)
Author : c1ccccc1
Score : 82 points
Date : 2024-12-16 03:39 UTC (19 hours ago)
(HTM) web link (johncarlosbaez.wordpress.com)
(TXT) w3m dump (johncarlosbaez.wordpress.com)
| bsaul wrote:
| i'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 wrote:
| Newtonian 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 wrote:
| It 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-
| mat...
| seanhunter wrote:
| Newtonian 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/esdif
| f/e...
|
| [3] Even though that was also actually invented by Euler
| apparently.
| pif wrote:
| > Everytime i manipulate dx i feel like walking on a minefield.
|
| Embrace the minefield, love the minefield!
| Signed a physicist
| xeonmc wrote:
| SICM?
| Y_Y wrote:
| If 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)
|
| (And obviously Functional Differential Geometry by the
| authors of SICM)
| leoc wrote:
| Certainly Gerry Sussman's frustrations with ambiguous
| notation were a big reason for his decision to create SICM!
| https://youtu.be/arMH5GjBwUQ?t=236
| TachyonicBytes wrote:
| You can always try infinitesimal analysis[1]
|
| [1] https://people.math.wisc.edu/~hkeisler/calc.html
| Chris2048 wrote:
| 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.
|
| 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 wrote:
| > 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 wrote:
| Would you mind giving us the titles of those books?
| Chris2048 wrote:
| They 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
| /n8...
|
| On that page the 'h' term is the infitesimal, as in
| d(x^2) / dx = 2x + h
|
| Though I prefer something like 'Dx' to make the link to x
| more explicit. Would love to see a more modern book on
| the topic.
| azalemeth wrote:
| Or, for the Knuth-included, _surreal_ numbers: https://people
| .math.harvard.edu/~knill/teaching/mathe320_201...
|
| (Seriously though, learn to love the minefield. ~~~~Another
| physicist)
| Qem wrote:
| Keisler book is excellent.
| BobaFloutist wrote:
| The 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 wrote:
| That's xkcd 927,a mnemonic for that is 3^2 * 100 + 3^3
| semi-extrinsic wrote:
| The 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).
| yccs27 wrote:
| This is the crucial insight. The last proof in TFA writes it in
| differential form notation, where "dx at constant y" is dx[?]dy
| = -dy[?]dx.
| quibono wrote:
| I think the video linked in the post does this, except for the
| 2-dimensional case.
| slooonz wrote:
| I 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 wrote:
| Imagine 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
|
| [2] https://en.wikipedia.org/wiki/File:Partial_func_eg.svg
| otisv wrote:
| I think you mean z=g(y)=f(3,y) instead of g(x)
| slooonz wrote:
| That'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 ?
| msravi wrote:
| > things like "partial derivative of pressure with respect
| to volume" and add "at constant temperature"
|
| They're the same thing, isn't it? Except that with add the
| "at constant temperature" addendum, you're just making
| explicit the other variable(s) that can potentially be
| varied. Without it, it just means all other variables,
| whatever they may be, are constant.
|
| But if something depended on both temperature and some
| other quantity X, and you said "partial derivative of
| pressure with respect to volume at constant temperature,"
| that would be sort-of misleading because you're only
| exlicitly mentioning one of the other two variables -
| rather, you should say "at constant temperature and X" or
| not mention either of them.
| bubblyworld wrote:
| They aren't the same thing since the first is strictly
| speaking not well defined - see my answer to the OP. I
| think the problem is that physicists use the same letter,
| say U, to denote multiple different mathematical
| functions depending on the context. The "holding XXX
| constant" thing serves to tell you which function you're
| dealing with formally.
| bubblyworld wrote:
| I assume you're talking about thermodynamics - this comes down
| to a slight abuse of notation. For an ideal gas, say, you can
| express various state functions like the internal energy in
| various different ways. You can do it in terms of pressure P
| and volume V to get U ~ PV, for instance.
|
| Or you could do it in terms of temperature T and pressure, for
| instance, to obtain U ~ T (in this case there's no dependence
| on pressure).
|
| The ideal gas laws let you transform between these choices. But
| the point is that the same physical quantity, U, has multiple
| mathematical functions underlying it - depending on which pair
| you choose to describe it with!
|
| To disambiguate this physicists write stuff like (dU/dP)_T,
| which means "partial derivative of U wrt P, where we use the
| expression for U in terms of P and T". Note that this is not
| the same as (dU/dP)_V, despite the fact that it superficially
| looks like the same derivative! The former is 0 and the latter
| is ~V, which you can compute from the expressions I gave above.
|
| The mistake is thinking that U is a single function of many
| independent variables P, T, S, V, etc. Actually these variables
| all depend on each other! So there are many possible functions
| corresponding to U in a formal sense, which is something people
| gloss over because U is a single physical quantity and it's
| convenient to use a single letter to denote it.
|
| Maybe it would make more sense to use notation like U(T, P) and
| U(P, V) to make it clear that these are different functions, if
| you wanted to be super explicit.
| slooonz wrote:
| Yes, exactly the explanation I looked for, thank you !
| AnimalMuppet wrote:
| > The mistake is thinking that U is a single function of many
| independent variables P, T, S, V, etc. Actually these
| variables all depend on each other!
|
| So, in vector space terms, we have different bases for
| describing U in, but not that many independent variables.
|
| If U is a function of x and y, but x and y are not
| orthogonal, then I can't treat dU/dx and dU/dy as
| independent, even for partial derivatives, because x and y
| aren't really independent.
| Y_Y wrote:
| You're not, in general, just working in a vector space but
| on a manifold whose coordinates are your extensive
| variables. It's only linear locally, in the (co-)tangent
| space where you're doing calculus.
|
| See e.g. https://arxiv.org/pdf/1811.04227
|
| Or this Physics SE discussion:
| https://physics.stackexchange.com/questions/388318/how-
| exact...
| movpasd wrote:
| Here's a geometric way of looking at it. I'll start with a
| summary, and then give a formal-ish description if that's more
| your jam.
|
| ---
|
| The fundamental issue is physicists use the same symbol for the
| physical, measurable quantity, and the function relating it to
| other quantities. To be clear, that isn't a criticism: it's a
| notational necessity (there are too many quantities to assign
| distinct symbols for each function). But that makes the
| semantics muddled.
|
| However, there is also a lack of clarity about the semantics of
| "quantities". I think it is best to think of quantities as
| functions over an underlying state space. Functional
| relationships _between_ the quantities can then be
| reconstructed from those quantities, subject to uniqueness
| conditions.
|
| This gives a more natural interpretation for the derivatives.
| It highlights that an expression like S(U, N, V) doesn't imply
| S _is_ the function, just that it's associated to it, and that
| S as a quantity could be associated with other functions.
|
| ---
|
| The state space S has the structure of a differential manifold,
| diffeomorphic to R^n [0].
|
| A quantity -- what in thermodynamics we might call a "state
| variable" -- is a smooth real-valued function on S.
|
| An diffeomorphism between S and R^n is a co-ordinate system.
| Its components form the co-ordinates. Intuitively, any
| collection of quantities X = (X_1, ..., X_n) which uniquely
| labels all points in S is a co-ordinate system, which is the
| same thing as saying that it's invertible. [1]
|
| Given such a co-ordinate system, any quantity Y can naturally
| be associated with a function f_Y : R^n -> R, defined by
| f_Y(x_1, ..., x_n) := Y(X^-1(x_1, ..., x_n)). In other words,
| this is the co-ordinate representation of Y. In physics, we
| would usually write that, as an abuse of notation: Y = Y(X_1,
| ..., X_n).
|
| This leads to the definition of the partial derivative holding
| some quantities constant: you map the "held constant"
| quantities and the quantity in the denominator to the
| appropriate co-ordinate system, then take the derivative of
| f_Y, giving you a function which can then be mapped back to a
| quantity.
|
| In that process, you have to make sure that the held constant
| quantities and the denominator quantity form a co-ordinate
| system. A lot of thermodynamic functions are posited to obey
| monotonicity/convexity properties, and this is why. It might be
| also possible to find a more permissive definition that uses
| multi-valued functions, similar to how Riemann surfaces are
| used in complex analysis.
|
| To do that we'd probably want to be a bit more general and
| allow for "partial co-ordinate systems", which might also be
| useful for cases involving composite systems. Any collection of
| quantities (Y, X_1, ..., X_n) can be naturally associated with
| a relation [2], where (y, x_1, ..., x_n) is in the relation if
| there exists a point s in S such that (Y(s), X_1(s), ...,
| X_n(s)) = (y, x_1, ..., x_n). You can promote that to a
| function if it satisfies a uniqueness condition.
|
| I think it is also possible to give a metric (Riemannian)
| structure on the manifold in a way compatible with the Second
| Law. I remember skimming through some papers on the topic, but
| didn't look in enough detail.
|
| ---
|
| [0] Or half of R^n, or a quadrant maybe.
|
| [1] The "diffeomorphism" definition also adds the condition
| that the inverse be smooth.
|
| [2] Incidentally, same sense of "relation" that leads to the
| "relational data model"!
| binary132 wrote:
| It just means fix that variable
| jwmerrill wrote:
| For problems in the plane, it's natural to pick two coordinate
| functions and treat other quantities as functions of these. For
| example, you might pick x and y, or r and th, or the distances
| from two different points, or...
|
| In thermodynamics, there often isn't really one "best" choice
| of two coordinate functions among the many possibilities
| (pressure, temperature, volume, energy, entropy... these are
| the must common but you could use arbitrarily many others in
| principle), and it's natural to switch between these
| coordinates even within a single problem.
|
| Coming back to the more familiar x, y, r, and th, you can
| visualize these 4 coordinate functions by plotting iso-contours
| for each of them in the plane. Holding one of these coordinate
| functions constant picks out a curve (its iso-contour) through
| a given point. Derivatives involving the other coordinates
| holding that coordinate constant are ratios of changes in the
| other coordinates along this iso-contour.
|
| For example, you can think of evaluating dr/dx along a curve of
| constant y or along a curve of constant th, and these are
| different.
|
| I first really understood this way of thinking from an
| unpublished book chapter of Jaynes [1]. Gibbs "Graphical
| Methods In The Thermodynamics of Fluids" [2] is also a very
| interesting discussion of different ways of representing
| thermodynamic processes by diagrams in the plane. His companion
| paper, "A method of geometrical representation of the
| thermodynamic properties of substances by means of surfaces"
| describes an alternative representation as a surface embedded
| in a larger space, and these two different pictures are
| complimentary and both very useful.
|
| [1] https://bayes.wustl.edu/etj/thermo/stat.mech.1.pdf
|
| [2] https://www3.nd.edu/~powers/ame.20231/gibbs1873a.pdf
| rob_c wrote:
| Am 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 wrote:
| Why 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 wrote:
| Well the fact that they're often written as fractions might be
| one reason...
| seanhunter wrote:
| My 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
|
| [1] https://mathworld.wolfram.com/Differentialk-Form.html
| jasomill wrote:
| A 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
|
| [2] https://mathworld.wolfram.com/CotangentBundle.html
|
| [3] https://mathworld.wolfram.com/FiberBundle.html
| brooke2k wrote:
| they 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"
| marcosdumay wrote:
| If you are careful to represent them on the right set of
| variables, and apply them on the right points (what the example
| on the article obviously doesn't do), they pretty much behave
| exactly like fractions.
|
| There are many areas of mathematics that spun from this.
| xorvoid wrote:
| I'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.
| nyeah wrote:
| Ordinary 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.
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