Find a function f such that f(xy)≠f(yx) at (0,0).

[hivesaeed4]

Mixed Partials:
Find a function f such that f(xy)≠f(yx) at (0,0).
Help?

 

[brydustin]

Mixed Partials:
Find a function f such that f(xy)≠f(yx) at (0,0).
Help?

What do you mean by f(xy)? You didn't use any commas and surely we're not supposed to assume you mean the product of xy vs yx?

 

[D H]

He means
$$\frac{\partial^2 f(x,y)}{\partial x \partial y} \ne \frac{\partial^2 f(x,y)}{\partial y \partial x}$$

 

[I like Serena]

Hi hivesaeed4!

Obviously any regular function of x and y won't work, since any function that is continuously differentiable two times wil have f_xy(0,0)=f_yx(0,0).


Did you try anything?
What kind of functions would you think of?
Can you give an example?
If you show some work, some stuff that you tried, we can help you further.


To help you on the way, you will need to define a function that is different based on conditions on its domain.
This also means you have to go back to the actual definition of the derivative to see how it works out.

To get to for instance f_xy(0,0), you need:
$$f_x(0,y) = \lim_{h \to 0} \frac{f(h,y)-f(0,y)}{h}$$
$$f_{xy}(0,0) = \lim_{h \to 0} \frac{f_x(0,h)-f_x(0,0)}{h}$$

 

[hivesaeed4]

Every function that I think of either gor=es to zero and/or f(xy) becomes equivalent to f(yx). I was thinking of using a function which had different powers of the two variables but then if the powers exceed 1 then always after the mixed partials are evaluated the function goes to zero cause one of the variables was left behind.
Any pointers in the right direction?

 

[DonAntonio]

Every function that I think of either gor=es to zero and/or f(xy) becomes equivalent to f(yx). I was thinking of using a function which had different powers of the two variables but then if the powers exceed 1 then always after the mixed partials are evaluated the function goes to zero cause one of the variables was left behind.
Any pointers in the right direction?

Check this: http://www.math.uconn.edu/~leibowitz/math2110f09/mixedpartials.pdf

DonAntonio

 

[hivesaeed4]

Thanks.

 

[brydustin]

Here's another example: http://www.math.byu.edu/~klkuttle/AdvancedCalculusMV.pdf

Read pages 133-135.