There has been this question that had been bothering me for a while and I could not find a satisfying answer on the internet or any of the books even though it is not very complex.
i) Its because if I have to find a third order polynomium approximation using taylor series for a 2 variable function, then is it correct to write that the third term will look something like this,
$$ ... + \frac{1}{3!}[f_{xxx}(x_0,y_0)(x-a)^3 + 6f_{xxy}(x_0,y_0)(x-a)(y-b)+f_{yyy}(x_0,y_0)(y-b)^3] + .... $$
I was a bit unsure about the middle part.
ii) About the hessian matrix, how would I write a hessian matrix if I have to make one for a third order like the one above. I know that for second order it looks like,
$$H_f(x,y) = \left(\begin{array}{cccc} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{array}\right)$$
Thank You :)