This discussion is taken from P. Topping's "Lectures on the Ricci Flow". The author defined the differential operator $L:C^\infty (M) \to C^\infty (M)$ as: $$ L(u) = a_{ij}\partial_i \partial_j u +b_i \partial_i u + c u $$ It's not written anywhere, but I guess one should sum over $i,j$ in this definition.
The author then presents the concept of a principal symbol of a differential operator as follows:
Define the principal symbol $\sigma(L) : T^*M \to \mathbb{R}$ by: $$ \sigma (L) (x,\xi) = a_{ij}(x)\xi_i \xi_j $$ ... An equivalent definition would be the following: Given $(x,\xi)\in T^*M$ and $\phi,f:M\to\mathbb{R}$ smooth with $d\phi(x)=\xi$ define: $$ \sigma (L) (x,\xi)f(x) = \lim_{s\to\infty} s^{-2} e^{-s\phi(x)}L(e^{s\phi}f)(x) $$
A proof that: $$ \lim_{s\to\infty} s^{-2} e^{-s\phi(x)}L(e^{s\phi}f)(x) = a_{ij}(x)\xi_i \xi_j \textbf{f(x)} $$ Follows. Note the bold trailing $f$.
I can see why those definitions are similar, but it seems like their domains mismatch - The second definition seems to be of type: $$\sigma(L):T^*M \times C^\infty (M) \to \mathbb{R}$$
Is there a type change of some sort I should be doing to get the equivalence ? Something else perhaps?