## anonymous one year ago ques...

1. anonymous

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

2. anonymous

@Loser66

3. Loser66

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

4. anonymous

What's that mean ??

5. Loser66

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

6. anonymous

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

7. Loser66

kind of

8. Loser66

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 $$f"_{y,x}= 5$$

9. Loser66

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.

10. IrishBoy123

.

11. anonymous

hmm kk

12. IrishBoy123

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.