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could any one just tell me in a short what is the geometric idea of this theorem?

"let $u:\Omega\subseteq\mathbb{R}^2\rightarrow \mathbb {R}$,$p=(x_0,y_0)\in\Omega$, $u_x,u_y$ exist at every point in a neighborhood of $p$ and continous at $p$ Then for sufficiently small $s,t$ in $\mathbb{R}$,

$$u(x_0+s,y_0+t)-u(x_0,y_0)=su_x(x_0+s^*,y_0+t^*)+tu_y(x_0,y_0+t^*)$$ with $|s^*|<|s|,|t^*|<|t|$

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    Isn't this just a generalization of the Mean Value Theorem?2012-08-24
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    how mow khau?...2012-08-24
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    You go from $(x,y)$ to $(x,y+t)$ applying the mean value theorem on $t$. Then you go from $(x,y+t)$ to $(x+s,y+t)$ applying the mean value theorem on $s$. The latter doesn't quite fit because $u_x$ is evaluated at $(x+s^*,y+t^*)$ and not $(x+s^*,y+t)$; are you sure that's not a typo?2012-08-24

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