How Can You Visualize Second Order Partial Derivatives?

[yitriana]

How do you visualize a second order partial derivative with respect to x and then y?

fxy or fyx?(same thing, but how to visualize)

 

[tiny-tim]

Hi yitriana!

(try using the X₂ tag just above the Reply box )

How do you visualize a second order partial derivative with respect to x and then y?

fxy or fyx?(same thing, but how to visualize)

You can visualise f_x as a slope (on an x y z "hillside" graph), and f_xx as a curvature.

f_xy would be 1/2 of the difference between the curvatures in the two "diagonal" directions, f_uu and f_vv where u = (x+y)/√2 and v = (x-y)/√2

 

[Hurkyl]

Another thing you might do is look at a "standard" representative.

For example, the function f(x,y) = xy has:

f(0,0) = 0
f_x(0,0) = 0
f_y(0,0) = 0
f_xx(0,0) = 0
f_xy(0,0) = 1
f_yy(0,0) = 0

and all other derivatives are zero.

So, in some sense, the behavior of this function near (0,0) sets the standard for what the effect of that derivative "looks like".

 

[yitriana]

Thank you for the replies. tiny-tim's explanation was especially helpful.