Taylor series third order approximation

Question

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 :)

Answer 1

i) Looks like you want: $\frac{1}{6} \left(3 (x-a)^2 (y-b) f_{\text{xxy}}\left(x_0,y_0\right)+3 (x-a) (y-b)^2 f_{\text{xyy}}\left(x_0,y_0\right)+(x-a)^3 f_{\text{xxx}}\left(x_0,y_0\right)+(y-b)^3 f_{\text{yyy}}\left(x_0,y_0\right)\right)$

ii) There's no third order "hessian matrix"

1st order derivative -> 2D vector

2nd order derivative -> 2x2 symmetric matrix

3rd order derivative -> 2x2x2 symmetric tensor.

Check wiki link.