Consider the following
We showed that Lie point symmetries are useful for solving ODE's and PDE's. But how do we find them the first place?
5.4.1 Trivial Case Lets try to find Lie point symmetries of an 0th order ODE $$ \Delta[x, u]=0 . $$ By definition $g^{\varepsilon}:(x, u) \mapsto(\tilde{x}, \tilde{u})$ is a Lie point symmetry if $$ \Delta[x, u]=0 \quad \Longrightarrow \Delta[\tilde{x}, \tilde{u}]=\Delta\left[g^{\varepsilon}(x, u)\right]=0 . $$ Question: Can we reduce this to a statement about the generator of $g^{\varepsilon}$ ? The answer is yes, but will need to assume that $\Delta$ is of maximal rank.
Defn (Maximal Rank) The operator $\Delta$ is of maximal rank if the matrix of derivatives $$ \frac{\partial \Delta_j}{\partial y_i} $$ is of maximal rank, where the $y_i$ runs over $x, u$, and in general all coordinates.
The definition makes no sense to me. The text gives $$ \Delta[x, u] = x^{2}. $$ as an operator which does not have maximal rank. The issue for me is that this is not a vector, so how can I take its $i$-th component?
Question: Could someone clarify here what the partial derivatives mean with this indentation?