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Are there any geometric interpretation for the second partial derivative? i.e.

$$f_{xy} = \frac {\partial^2 f} {\partial x \partial y}$$

In particular, I'm trying to understand the determinant from second partial derivative test for determining whether a critical point is a minima/maxima/saddle points:

$$D(a, b) = f_{xx}(a,b) f_{yy}(a,b) - f_{xy}(a,b)^2$$

I have no trouble understanding $f_{xx}(x,y)$ and $f_{yy}(x,y)$ as the of measure of concavity/convexity of f in the direction of x and y axis. But what does $f_{xy}(x,y)$ means?

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    there is a taylor theorem (polynomial approximation) for functions of several variables, here $\mathbb{R}^2\to\mathbb{R}$. the second derivative test looks at what kind of 2nd degree polynomial (in two variable) approximates the function. if it is a paraboloid or hyperboloid you can infer max/min/saddle properties but if it is flat in some direction it is inconclusive.2011-03-28
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    $\frac{\partial}{\partial y}f$ tells you how $f$ is changing "in the $y$-direction", but that change will generally depend on the $x$-position (think of a grid tiling the plane; the partial tells you how things are changing in the vertical direction, but the change depends on which "column" you are in). If you think about it as the slope of the "tangent in the $y$-direction", then as you move the point in the $x$-direction this slope changes as well; the change in that slope is given by $\frac{\partial^2}{\partial x\partial y}$.2011-03-28
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    Seems you're asking a subquestion of what you want--- not how to interpret mixed partials, but why the sign of $D(a,b)$ can give the nature of a saddle point. For this, do elementary analytic geometry on the graph of a function $Ax^2 + By^2 + Cxy$ at $(0,0)$ (add $Ex + Fy + G$, and at $(a,b)$, if you don't see how to reduce to this case). What conditions on $A,B,C$ do you get bowl that opens up, bowl that opens down, or saddle? Once you "get" this, you "get" all $f$, by the Taylor theorem. (Personally, I understand this via the algebra, not "geometric understanding" of $f_{xy}$.)2011-03-28
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    @anon: yes, it's a subquestion of what I want to ultimately understand; I'm trying to understand why/how the determinant works from a geometric perspective, and to do that, I believe I need to understand the mixed derivative first.2011-03-29
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    http://www.math.harvard.edu/archive/21a_fall_08/exhibits/fxy/index.html2012-03-07

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