Question

ques...

Answer 1

\[\frac{\partial^2f(x,y)}{\partial x \partial y}=\frac{\partial^2f(x,y)}{\partial y \partial x}\]
Is this always true???

Answer 2

@Loser66

Answer 3

NOPPPPPe. It's true IF AND ONLY IF f(x,y) is a symmetric one.

Answer 4

What's that mean ??

Answer 5

like \(f(x,y)= x^2 + y^2 + x +y\) what happens to x, it happens to y.
if \(f(x,y) = x^2 + y \) then partial x, y is different from partial y, x

Answer 6

wait symmetric as in do u mean
f(x,y)=f(y,x)??

Answer 7

kind of

Answer 8

Other example: \(f(x,y) = 2x^2 + 5xy + 2y^2\)
partial w.r.t x = \(f'_x= 4x +5y\), then w.r.t y \(f"_{x,y}= 5\)
partial w.r.t.y = \(f'_y =4y +5x\) then, w.r.t x \(f"_{y,x}= 5\)

Answer 9

to higher degree of a function, the letters switch to each other, but the answer are the same
like \(f(x,y) = x^3+ 20x^2y + y^3\), it is a symmetric one, to find \(f"_{x,y}~~f"_{y,x}\), you can save time by doing just one, then switch the letters.

Answer 10

.

Answer 11

hmm kk

Answer 12

according to the Gospel of Mary (Boas : Mathematical Methods in Physical Sciences [p190, 3rd Edition, if you have access] ), so long as \(f_x, f_y, f_{xx} and f_{yy}\) are continuous, then \(f_{xy} = f_{yx}\)

here's an example of an exception:
http://www.math.tamu.edu/~tvogel/gallery/node18.html

and if you google something like "mixed partials not equal" i think you will find more stuff.

my sense FWIW is that if you are doing physical/applied stuff you will be more than aware of the discontinuity and can probably even find a way around it.