From pages 418-419 of PDE Evans...
We begin by considering a general linear second-order PDE in two variables $$\tag{82} \sum_{i,j=1}^2 a^{ij} u_{x_i x_j} + \sum_{i=1}^2 b^i u_{x_i} + cu = 0,$$ where the coefficients $a^{ij},b^i,c (i,j=1,2)$, with $a^{ij}=a^{ji}$, and the unknown $u$ are functions of the two variables $x_1$ and $x_2$ in some region $U \subset \mathbb{R}^2$.
The book goes on to say
Let us set \begin{cases}y_1 = \Phi^1 (x_1,x_2) \\ y_2 = \Phi^2 (x_1,x_2) \tag{83} \end{cases} for some appropriate function $\mathbf{\Phi} = (\Phi^1, \Phi^2)$. To investigate this possibility let us now write $$u(x)=v(\mathbf{\Phi}(x)). \tag{84}$$ That is, we define $v(y) := u(\mathbf{\Psi}(y))$, where $\mathbf{\Psi}=\mathbf{\Phi}^{-1}$.
$\quad$From $\text{(84)}$, we compute \begin{cases}u_{x_i} = \sum_{k=1}^2 v_{y_k} \Phi_{x_i}^k \\ u_{x_i x_j}=\sum_{k,l=1}^2 v_{y_k y_l} \Phi_{x_i}^k \Phi_{x_j}^l + \sum_{k=1}^2 v_{y_k} \Phi_{x_i x_j}^k \end{cases} for $i,j=1,2$. Substituting into $\text{(82)}$, we discover that $v$ solves the PDE $$\sum_{k,l=1}^n \tilde{a}^{kl} v_{y_k y_l} + \cdots = 0, \tag{85}$$ for $$\tag{86} \tilde{a}^{kl} := \sum_{i,j=1}^2 a^{ij} \Phi_{x_i}^k \Phi_{x_j}^l \quad (k,l=1,2),$$ where the dots in $\text{(85)}$ represent terms of lower order.
Okay, when I insert both $\text{(84)}$ and \begin{cases}u_{x_i} = \sum_{k=1}^2 v_{y_k} \Phi_{x_i}^k \\ u_{x_i x_j}=\sum_{k,l=1}^2 v_{y_k y_l} \Phi_{x_i}^k \Phi_{x_j}^l + \sum_{k=1}^2 v_{y_k} \Phi_{x_i x_j}^k \end{cases} into the equation in $\text{(82)}$, I get $$\sum_{i,j=1}^2 a^{ij} \left(\sum_{k,l=1}^2 v_{yk y_l} \Phi_{x_i}^k \Phi_{x_j}^l + \sum_{k=1}^2 v_{y_k} \Phi_{x_i x_j}^k \right) + \sum_{i=1}^2 b^i \left(\sum_{k=1}^2 v_{y_k} \Phi_{x_i}^k \right) + cv = 0$$ or, when applying $\text{(86)}$, $$\sum_{k,l=1}^2 \tilde{a}^{kl} v_{y_k y_l} + \sum_{k=1}^2 \tilde{a}^{kl} v_{y_k} + \sum_{i,k=1}^2 b^i v_{y_k} \Phi_{x_i}^k + cv = 0,$$ and I am unsure of how to arrive at $\text{(85)}$. Specifically, how do they even get the $n$ in the notation $\sum_{k,l=1}^n$ in $\text{(85)}$ when we've only been dealing with only the two variables $(x_1,x_2) \in \mathbb{R}^2$, which results in dealing with the notation $\sum_{k,l=1}^2$?