I have a function of two variables $f(x,y)$. This function has a Taylor series, $$ f(x,y) = \sum_{n=0}^\infty\sum_{m=0}^\infty c_{nm}x^n y^m. $$ I define $g(x,y)$ by the Taylor series of $f(x,y)$ truncated as follows $$ g(x,y) = \sum_{n=0}^N c_{n0}x^n + \sum_{m=0}^M c_{0m}y^m, $$ i.e. $g(x,y)$ is $f(x,y)$ with all cross-terms discarded and with the sums truncated at $N$ and $M$.
My question is: what notation should I use to relate $g(x,y)$ and $f(x,y)$? Can I use big $\mathcal{O}$ notation e.g. $$ f(x,y) = g(x,y) + \mathcal{O}(x^{N+1} + y^{M+1}+xy)\ ? $$ Any clarification on this would be really appreciated. Thanks all!