# Notation for expansion of multivariable functions

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be some analytic function. I want to say something like, as $x, y \rightarrow 0$, the taylor expansion looks like: $$f(x,y) = a x^2 + b y^2 + \text{(monomials of degree larger than 3)}$$ What's the cleanest way to say this? I was thinking for example, as $x,y \rightarrow 0$ $$f(x,y) = a x^2 + b y^2 + O\left( \sqrt{x^2 + y^2}^3 \right)$$ meaning to say, terms that are bounded by some constant times $\sqrt{x^2+y^2}^3$ as $x,y \rightarrow 0$ but I am not sure if this standard notation.