Question

Help on notation.

Answer 1

What is the order of differentiation in \(\dfrac{\partial^2 f}{\partial y\partial x}\)? How about \(\left[D_{y,x}f\right](x,y)\)?

@Kainui

Answer 2

@ikram002p Simple question and need answer quick lol.

Answer 3

\[\frac{\partial^2 f}{\partial y\partial x} = \frac{\partial}{\partial y} \left( \frac{\partial}{\partial x} f\right)\]
That is how it should be read, like you would write \((f \circ g )(x) = f(g(x))\)

This on the other hand:
\(\left[D_{y,x}f\right](x,y)\)

I've never seen this notation before so I can't tell you. I've seen stuff like this before however:
\[f_{,y} = \frac{\partial}{\partial y} f\]

Answer 4

I am just reading a proof of Schwarz theorem which states that if the mixed derivative is continuous at that point, the order of differentiation doesn't matter. Totally disastrous lol.

Answer 5

Haha yeah, there are some problems with this and this ends up being a pretty important concept later too. If your partial derivatives aren't interchangeable then it tells you the curvature of the space which is pretty cool.

Answer 6

How about \(f_{xy}\)?

Answer 7

The convention is \(f_{xy} = (f_x)_y\) haha. Messed up, I know.

Answer 8

So \(f_{xy}=\dfrac{\partial^2 f}{\partial y \partial x}\)? What....??????

Answer 9

Exactly

Answer 10

I don't know what to say now.........

Answer 11

I better play Sudoku and kill myself.

Answer 12

rip https://upload.wikimedia.org/math/5/a/f/5afe802ca198aed9cedadaa8579a7a08.png (also idk what the hell the weird ' notation is on the far left, never seen ANYONE use that crap)

Answer 13

Exactly. What crap is that?

Answer 14

idk, but you can see that pic I linked is backwards of what I said too smh

Answer 15

I don't even know anymore, I have never seen it the other way. To be fair though, most of the time partial derivatives commute so it doesn't matter lol

Answer 16

I think we could safely ignore that crap. How do you express \(\dfrac{\partial^2 f}{\partial x \partial y} \neq \dfrac{\partial^2 f}{\partial y \partial x}\) in that notation? \(f'_\prime \neq f'_\prime\)????????

Answer 17

weird the wikipedia page is bunk I think cause going to a random google page some guy says exactly what I just said https://www.physicsforums.com/threads/subscripts-in-partial-derivative-notation.678870/

Answer 18

haha yeah you can't. Well in that notation you're right it necessarily would mean \(f_{xy}=f_{yx}\) good catch. Well ok I feel less doubt now

Answer 19

https://www.math.hmc.edu/calculus/tutorials/partialdifferentiation/

Yeah they say the same thing as I'm saying here too good

Answer 20

How about a function of three variables? What is this nonsense!!??????