Let $X\subset\mathbb{P}^n$ be an hypersurface given by the vanishing of a polynomial $F\in k[x_0,...,x_n]_d$. Let $Y\subset X$ be a subvariety. Then $X$ has multiplicity $m$ along $Y$ if all the partial derivatives of order $m-1$ of $F$ vanish on $Y$ but there is at least one partial derivative of $F$ of order $m$ not vanishing along $Y$.

I am wondering if there is a criterion like this for higher codimension varieties.

More precisely, assume that $X\subset\mathbb{P}^n$ has codimension greater that one, and let $I = (F_1,...,F_r)$ be its homogeneous ideal. Is it true that $X$ has multiplicity $m$ along $Y$ if all the partial derivatives of order $m-1$ of $F_1,...,F_r$ vanish on $Y$ but there is at least one partial derivative of $F_i$, for some $i$, of order $m$ not vanishing along $Y$ ?