Geometric interpretation for d²f/dxdy

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Discussion Overview

The discussion revolves around the geometric interpretation of the mixed partial derivative ##f_{xy}(x_0, y_0)## in the context of a double integral. Participants explore how this derivative relates to changes in a function over an area.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant suggests that the geometric interpretation of ##f_{xy}(x_0, y_0)## could be the rate of change of z with respect to area, expressed as ##dz/dA## at the point (x0, y0).
  • Another participant reiterates the idea of rate of change of z with area but seeks further clarification on the geometric interpretation.
  • A different perspective is introduced, where a participant proposes that if A represents a shaded area, then ##f(x,y) = dA/dR##, indicating that the derivative ##df/dR## measures curvature, suggesting that when it is zero, the area A is flat.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the geometric interpretation of ##f_{xy}(x_0, y_0##), with multiple competing views and interpretations presented.

Contextual Notes

There are unresolved assumptions regarding the definitions of the variables involved and the specific context of the area and curvature being discussed.

Jhenrique
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If the following integral:
$$\\ \iint\limits_{a\;c}^{b\;d} f(x,y) dxdy$$ represents:

attachment.php?attachmentid=70578&stc=1&d=1402650671.png


So which is the geometric interpretation for ##f_{xy}(x_0, y_0)## ?
 

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Would it be the rate of change of z with area, dz/dA at x0, y0?
 
Jilang said:
Would it be the rate of change of z with area, dz/dA at x0, y0?

Yeah! But which is the geometric interpretation?
 
I am taking my last guess back. If A is the shaded area
f(x,y) = dA/dR
So df/dR is a measure of the curvature. When it is zero the area A is flat.
 

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