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\[ f(x,y)=f(x_0,y_0)+(x-x_0)\frac{\partial f}{\partial x}+(y-y_0)\frac{\partial f}{\partial y}+\frac{1}{2!} ( (x-x_0)^2\frac{\partial^2 f}{\partial x^2} \] \[ +2(y-y_0)(x-x_0)\frac{\partial^2f}{\partial xy}+(y-y_0)^2\frac{\partial^2 f}{\partial y^2})+... \] Why?

Mathematics
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I'm having trouble with getting the intuition behind the Taylor series when the function is multivariabled. I know that you can prove the Taylor series with integration by parts, but I'm not sure how you would use that here (if at all). I assume the answer to this question will be closely associated with the total differential of a multiplication variable to a high degree of approximation (I mean with dx^n dy^m lying about the place, where n>1 and/or m>1).
|dw:1351261479700:dw|
|dw:1351261612819:dw|

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So all coefficients define by m,n derivatives of f(x,y)
I understand what it is saying, but why is the second assumption allowable? That is the essence of my question.

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