For example, when we Taylor expand say $e^x$ about $x=0$, we would write
$$e^x = 1+x+\frac 12 x^2 + \frac 16 x^3 + \mathcal O(x^4) \qquad \qquad \text{as } x \rightarrow 0$$
with the use of the "Big-O" notation $\mathcal O$.
I was just wondering, is there a generalisation for this to "multiple variables"?
For example, if I Taylor expand $e^{x+y}$ about $(x,y) = (0,0)$, we get
$$e^{x+y} = 1 + x + y + \frac 12 x^2 + xy + \frac 12 y^2 + \text{higher order terms} \qquad \qquad \text{as } (x,y) \rightarrow (0,0)$$
where "higher order terms" in this case would refer to cubic or higher terms in $x,y$, namely $x^3$, $x^2y$, $xy^2$, $y^3$ or beyond. You can't just write something like $\mathcal O(x^3)$ or $\mathcal O((xy)^3)$ because there are terms like $x^2y$ that don't contain the $x^3$.
So...what's the correct way to express this?
You could write it as $\mathcal O(|x|^3 + |y|^3)$.