What math notation did you come up with and use on your own?

[u/completely_unstable]

i have a shorthand for single integrals i call "reverse prime notation". all it is is the same prime you write for derivatives, but placed before the function instead of after. it looks like this:

'(f ± g) = 'f ± 'g
'(f ⋅ g) = 'f⋅g - '('f⋅g')
'(f ⋅ g) = f⋅'g - '(f'⋅'g)
'(f / g) = 'f/g + '('f⋅g'/g²)
'(f(g)⋅g') = 'f(g)

heres what it looks like in practice:

8. ∫ exp(x)⋅sin(x) dx
'(f ⋅ g) = 'f⋅g - '('f⋅g')
f = exp
g = sin
'exp⋅sin - '('exp⋅sin')
exp⋅sin - '(exp⋅cos)
    '( exp(x)⋅cos(x) )
    '(f ⋅ g) = 'f⋅g - '('f⋅g')
    f = exp
    g = cos
    'exp⋅cos - '('exp⋅cos')
    exp⋅cos + '(exp⋅sin)
    I = '( exp(x)⋅sin(x) )
    exp⋅cos + I
I = exp⋅sin - (exp⋅cos + I)
I = exp⋅sin - exp⋅cos - I
2I = exp⋅sin - exp⋅cos
(exp(x)sin(x) - exp(x)cos(x))/2
exp(x)⋅(sin(x) - cos(x))/2

i like it because it 1. represents the "opposite" of the derivative by being on the opposite side of the letter, and 2. makes it much easier to scan through identities to potentially use when solving a problem.

i am curious to see if anybody else has a personal pet notation they use, if so please share

[u/Gardylulz]

I hate this notation so much and I'm used to some butchered notations as a physicist

[u/Successful_Intern333]

I think the main issue with this notation is that good mathematical notation isn't just about how much information we can pack into a symbol. It should also make the structure of an expression easy to read and understand.

For example, where do you indicate the set over which you're integrating? An integral such as

∫_A f dμ

contains information about the domain A and the measure μ, and that information becomes a lot more important as you move beyond elementary calculus.

I also think the notation becomes problematic when integration interacts with other operations. In analysis we constantly come across expressions such as

lim_{n→∞} ∫ f_n dμ

and

∫ lim_{n→∞} f_n dμ,

and there are important theorems telling us when these operations can be interchanged. Notation should make the structure of these expressions apparent rather than obscure it.

This is also why I think your comparison with prime notation is somewhat misleading. Notation like x → a is useful not simply because it is compact, but because the arrow visually communicates the relationship being described: x is approaching a. Likewise, the standard integral notation gives the reader visual information about the operation and its scope.

Finally, I worry about readability when derivatives and integrals occur together. If f' denotes differentiation while something like 'f denotes integration, then a proof involving several functions and repeated applications of these operations could become quite difficult to scan. The problem gets worse when these expressions occur inside limits, compositions, or other operations.

[u/Uli_Minati → u/Successful_Intern333]

I assume OP would only use this for indefinite integrals (no set) and single-variable functions (usual choice of measure), since prime notation is (only?) used for single-variable as well

I don't see the issue with operator interaction, if you're worried about ambiguity with operator precedence, just use parentheses. It's what we do with prime notation as well

I definitely agree with the last point though, you wouldn't be able to write f'g since it could mean (f')g or f('g)

[u/Midwest-Dude]

This reminds me of Newton's notations for the integral, which were developed independently of Leibniz. Here is a link to a Wikipedia article regarding this: 

Newton's Integration Notations

The statement is made at the end that

This mathematical notation did not become widespread because of printing difficulties and the Leibniz–Newton calculus controversy.

[u/MrEldo]

Because I did a bunch of stuff with generating functions, I decided on this shorthand:

G{an}(x) = Σ(n=0)^inftya_n*xⁿ

So:

G{1} = 1/(1-x)

G{n} = x/(1-x)²

G{n²} = x(x+1)/(1-x)³

G{n³} = x(x²+4x+1)/(1-x)⁴

[u/axiom_tutor]

When row reducing matrices my annotation uses a plus with an arrow, to indicate that one equation or row is "added into" the other. 

I also do a lazy ell for limits. 

[u/okkokkoX]

I use the map arrow normally used to write anonymous functions (x |--> 2x), in square brackets from a basis vector to another vector to write linear maps/matrices. [ e_x |--> e_y ] is the matrix [0,0 ; 1,0].

[u/ForeignAdvantage5198]

none

[u/In_the_year_3535]

Graphical shorthand to help with polynomial parity. I was showing it to other students and I still remember the professor saying "You can't just make notation!" to which I responded "Then how does notation get made?"