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About this particular case: why hasn't Euler's notation (https://en.wikipedia.org/wiki/Notation_for_differentiation#Euler.27s_notation https://en.wikipedia.org/

by nnq 9y ago

About this particular case: why hasn't Euler's notation (https://en.wikipedia.org/wiki/Notation_for_differentiation#Euler.27s_notation https://en.wikipedia.org/wiki/Notation_for_differentiation#E...) become the clear winner? I mean, its the only one that is both not confusing for beginners (because it doesn't trick them into the "cool, let's `simplify` the dx at the denominator with the next one" mindset) and it also translates easily to code (or other 1D encoding), like you can write "second derivative of f with respect to x" as D[f, x, 2] and "integral of a with respect to t" as "D[a, t, -1]". My point is mainly that there is nothing rational at all in how humans choose notations... without some obscure historic events, we'd probably still be using some derivative of roman numerals or maybe even sexagesimals! (https://en.wikipedia.org/wiki/Sexagesimal https://en.wikipedia.org/wiki/Sexagesimal) (I imagine the reason is because "highly performing" individuals have their own "internal" language to think in, so general language is just a for communication, so a political decision... unfortunately for education :()

tgb 9y ago
It's actually pretty common notation in a lot of fields, and mathematicians frequently switch between whichever of these notations is most convenient for a particular problem. Your point of course still stands for why isn't it the main one used in Calculus 1 courses, but it's common in differential equations courses in my experience (and PDE's often simplifies it further to just have D_x f as f_x). And many times where the "derivative" is not necessarily the "calc 1" derivative, you use notation similar to that, for example the covariant derivative [1] is sometimes written like that but with a 'nabla' instead of a 'D'. Some authors even use 'D' for covariant derivative along a path. [1] https://en.wikipedia.org/wiki/Covariant_derivative https://en.wikipedia.org/wiki/Covariant_derivative
bonoboTP 9y ago
I think a lot of my confusions when I first learned calculus would have been eliminated with a notation that clearly expressed that derivatives operate on whole functions, not on values. So the derivative of f(x) at x=0 is not some function of f(0), but it is derivative(f)(x). Also, even without derivatives, sometimes the expression f(x) refers to the whole function, sometimes just a particular value of it at a specific x. I don't know if higher-order functions are really that tough to teach/understand but I think it would actually simplify and demistify many things. Currently, of the widespread notations I like this one best: f(x0) = d/dx (sin(x)*cos(x)+x^2) | x=x0 (with the last part in subscript) I also miss variable scoping from math writing and it disturbs me that variable names often carry semantics, like p(x) and p(y) can be the probability density functions of random variables X and Y (so p is a different function depending on the name of the input variable that you substitute so it doesn't actually operate on real numbers, but (string, number) pairs). I'd prefer to explicitly mark the functions as p_X(x) and p_Y(y). Similar things come up a lot with differential equations where you don't really know whether something (like y or u) is supposed to be a function (of x or t) or "just" a variable. Despite the general opinion among laypeople that math notation is very precise and unambiguous, I find that it's often very sloppy and unless you already understand the context very well, it can easily be misleading. Math notation is somewhere between normal natural language and programming languages, and depending on the writer it may be closer to one or the other. One can argue that this is necessary for compactness.
- noir_lord 9y ago
  I've noticed that I understand what is going on much more when a function is written in code than in its mathematical form. A lot of that is familiarity but I don't think all of it is.
  - ashark 9y ago
    Code's easy for me (unless "mathy" in appearance like Haskell) but mathematical notation's always made me feel dyslexic. I'd love to see this beauty or clarity or whatever that people find in mathematics, but I've never caught even a hint of it. Seems like it needs a good IDE to make up for deficiencies in its language.
    - noir_lord 9y ago
      Have you see SageMath[0]? That allows you to use Python inside a mathematical environment and it's amazing. Saved my bacon a few times when I need to translate maths to code and I need to poke it a bit to get an understanding of how it works. [0]http://www.sagemath.org/ http://www.sagemath.org/
    - nnq 9y ago
      > Code's easy for me (unless "mathy" in appearance like Haskell) Haskell is not hard to read/understand because it's mathy. It's because of type inference combined with laziness - this leads people to do incredible fits of intellectual masturbation... Like "hey, the compiler can handle types for us, so lets type the shit out of things, let's even use them to mark side effects, types are fee" ...doesn't matter that types are mainly documentation and by not writing them or making astronaut types you're loosing the main benefit. And "hey, the language is lazy, that extra code containing an infinite loop but so mathy elegant, will never be fully executed so we can have it, it's free, doesn't impact performance"... doesn't matter you need to use 80% of your brainpower better spent otherwise to grok it. If it would be eager and without type inference it would've evolved into a much more readable and popular language and with a larger community.
  - vidarh 9y ago
slaymaker1907 9y ago
I like that the Euler notation makes it more clear that derivation is a higher order function.
marcosdumay 9y ago
It's not much more useful than Newton's. The greatest point of Leibniz' notation is that you can do those things such as simplifying dx, or moving dy to the other side of the equality. It is confusing to beginners, but it's very useful once you understand it.
Maken 9y ago
What I can't understand is why anybody still used Newton's notation. Also, at least for me Leibniz notation tends to make pen-and-paper symbol manipulation simpler.
deleted 9y ago
[deleted]

by nnq 9y ago

tgb 9y ago

It's actually pretty common notation in a lot of fields, and mathematicians frequently switch between whichever of these notations is most convenient for a particular problem. Your point of course still stands for why isn't it the main one used in Calculus 1 courses, but it's common in differential equations courses in my experience (and PDE's often simplifies it further to just have D_x f as f_x). And many times where the "derivative" is not necessarily the "calc 1" derivative, you use notation similar to that, for example the covariant derivative [1] is sometimes written like that but with a 'nabla' instead of a 'D'. Some authors even use 'D' for covariant derivative along a path. [1] https://en.wikipedia.org/wiki/Covariant_derivative https://en.wikipedia.org/wiki/Covariant_derivative

bonoboTP 9y ago

I think a lot of my confusions when I first learned calculus would have been eliminated with a notation that clearly expressed that derivatives operate on whole functions, not on values. So the derivative of f(x) at x=0 is not some function of f(0), but it is derivative(f)(x). Also, even without derivatives, sometimes the expression f(x) refers to the whole function, sometimes just a particular value of it at a specific x. I don't know if higher-order functions are really that tough to teach/understand but I think it would actually simplify and demistify many things. Currently, of the widespread notations I like this one best: f(x0) = d/dx (sin(x)*cos(x)+x^2) | x=x0 (with the last part in subscript) I also miss variable scoping from math writing and it disturbs me that variable names often carry semantics, like p(x) and p(y) can be the probability density functions of random variables X and Y (so p is a different function depending on the name of the input variable that you substitute so it doesn't actually operate on real numbers, but (string, number) pairs). I'd prefer to explicitly mark the functions as p_X(x) and p_Y(y). Similar things come up a lot with differential equations where you don't really know whether something (like y or u) is supposed to be a function (of x or t) or "just" a variable. Despite the general opinion among laypeople that math notation is very precise and unambiguous, I find that it's often very sloppy and unless you already understand the context very well, it can easily be misleading. Math notation is somewhere between normal natural language and programming languages, and depending on the writer it may be closer to one or the other. One can argue that this is necessary for compactness.

noir_lord 9y ago
I've noticed that I understand what is going on much more when a function is written in code than in its mathematical form. A lot of that is familiarity but I don't think all of it is.
- ashark 9y ago
  Code's easy for me (unless "mathy" in appearance like Haskell) but mathematical notation's always made me feel dyslexic. I'd love to see this beauty or clarity or whatever that people find in mathematics, but I've never caught even a hint of it. Seems like it needs a good IDE to make up for deficiencies in its language.
  - noir_lord 9y ago
    Have you see SageMath[0]? That allows you to use Python inside a mathematical environment and it's amazing. Saved my bacon a few times when I need to translate maths to code and I need to poke it a bit to get an understanding of how it works. [0]http://www.sagemath.org/ http://www.sagemath.org/
  - nnq 9y ago
    > Code's easy for me (unless "mathy" in appearance like Haskell) Haskell is not hard to read/understand because it's mathy. It's because of type inference combined with laziness - this leads people to do incredible fits of intellectual masturbation... Like "hey, the compiler can handle types for us, so lets type the shit out of things, let's even use them to mark side effects, types are fee" ...doesn't matter that types are mainly documentation and by not writing them or making astronaut types you're loosing the main benefit. And "hey, the language is lazy, that extra code containing an infinite loop but so mathy elegant, will never be fully executed so we can have it, it's free, doesn't impact performance"... doesn't matter you need to use 80% of your brainpower better spent otherwise to grok it. If it would be eager and without type inference it would've evolved into a much more readable and popular language and with a larger community.
- vidarh 9y ago

slaymaker1907 9y ago

I like that the Euler notation makes it more clear that derivation is a higher order function.

marcosdumay 9y ago

It's not much more useful than Newton's. The greatest point of Leibniz' notation is that you can do those things such as simplifying dx, or moving dy to the other side of the equality. It is confusing to beginners, but it's very useful once you understand it.

Maken 9y ago

What I can't understand is why anybody still used Newton's notation. Also, at least for me Leibniz notation tends to make pen-and-paper symbol manipulation simpler.

deleted 9y ago

[deleted]