Visualize partial derivatives?

  • Thread starter yitriana
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Main Question or Discussion Point

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)
 

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  • #2
tiny-tim
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Hi yitriana! :smile:

(try using the X2 tag just above the Reply box :wink:)
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 fx as a slope (on an x y z "hillside" graph), and fxx as a curvature.

fxy would be 1/2 of the difference between the curvatures in the two "diagonal" directions, fuu and fvv where u = (x+y)/√2 and v = (x-y)/√2 :wink:
 
  • #3
Hurkyl
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Another thing you might do is look at a "standard" representative.

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

f(0,0) = 0
fx(0,0) = 0
fy(0,0) = 0
fxx(0,0) = 0
fxy(0,0) = 1
fyy(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".
 
  • #4
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Thank you for the replies. tiny-tim's explanation was especially helpful.
 

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