## henpen 3 years ago $f(x,y)=f(x_0,y_0)+(x-x_0)\frac{\partial f}{\partial x}+(y-y_0)\frac{\partial f}{\partial y}+\frac{1}{2!} ( (x-x_0)^2\frac{\partial^2 f}{\partial x^2}$ $+2(y-y_0)(x-x_0)\frac{\partial^2f}{\partial xy}+(y-y_0)^2\frac{\partial^2 f}{\partial y^2})+...$ Why?

1. henpen

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).

2. mahmit2012

|dw:1351261479700:dw|

3. mahmit2012

|dw:1351261612819:dw|

4. mahmit2012

So all coefficients define by m,n derivatives of f(x,y)

5. henpen

I understand what it is saying, but why is the second assumption allowable? That is the essence of my question.