Third degree Taylor polynomial in two variables

Question

How does one find the third-degree Taylor polynomial of $f(x,y) = (x+y)^3$ at the points $(0,0)$ and $(1,1)$?

Many thanks

Answer 1

Hint: In general, $$\left.T_n(x,y)\right|_{\text{About $(x_0,y_0)$}}=\sum^n_{k=0}\sum^{n-k}_{l=0}\left.\dfrac{\partial^k\partial^lf}{\partial x^k\partial y^l}\right|_{(x_0,y_0)}\dfrac{(x-x_0)^k(y-y_0)^l}{k!\cdot l!}$$

Answer 2

Alternatively, and perhaps of greater difficulty, you can add zero or just multiply: $$ f(x,y) = (x+y)^3 = x^3+3x^2y+3xy^2+y^3 $$ and, $$ f(x,y) = ((x-1)+1+(y-1)+1)^3 = (\alpha + \beta + 2)^3 $$ expand and that gives you the expansion in $(x-1),(y-1)$. Since Taylor polynomials are unique when they exist, this recovers the same answer as indicated via the multivariate Taylor expansion in Sanath's excellent hint.