I am having difficulties figuring out how to write the full expansion of taylor's formula in 3 variables (x,y,t), for degrees 2, 3 and even higher degree terms.

It is for a master thesis project in computer vision.

The remainder term(s) is not necessary, but would be nice to have as well.

This is taylor's formula expansion for f(x) containing up to first degree terms only (with remainder $\displaystyle R_1$):

$\displaystyle f(b)=f(a)+f_x(a)(b-a) + R_1$

And, if we exchange some variables: $\displaystyle b=x+\Delta x$ and $\displaystyle a=x$, we have:

$\displaystyle f(x + \Delta x)=f(x)+f_x(x) \Delta x + R_1$

Using this notation instead, this is my taylor's formula expansion for f(x,y,t) up to first degree terms (partial derivatives in x, y, and t):

$\displaystyle f(x + \Delta x, y + \Delta y, t + \Delta t)=$

$\displaystyle f(x,y,t) + f_x(x,y,t) \Delta x + f_y(x,y,t) \Delta y + f_t(x,y,t) \Delta t + \text{remainder terms...}$

Now I need the expansion in all three variables, for higher degrees (second, third, perhaps four) How to do this? I have found functions for this, but these are using multi-index vectors for indexing, and unfortunately I have been unable to wrap my head around such notation.

As a side note, I currently expect mixed/crossed partials to be equal for function f(x,y,t) (which I do not know analytically).

Thanks,

Intel4004