One-parameter Infinitesimal transformation with m dependent variables

Question

I have been trying to find out how to write the one-parameter Infinitesimal transformation $Uf$ (Lie's continuous groups) with n dependent variables, so far without success.

Abraham Cohen, "An introduction to the Lie theory of one-parameter groups" gives the expression for infinitesimal transformation for one dependent variable $y$ and $n$ independent variables $x_1,x_2,...,x_n$:

Given is the point transformation

$$x_k^{\prime}=\phi_k(x_1,x_2,...,x_n,y,a), \mbox{ } k=1...n$$ $$y^{\prime}=\psi(x_1,x_2,...,x_n,y,a)$$

where $a$ is the parameter of the group.

Then, if $f(x_1,x_2,...,x_n,y)$ is a smooth function and has the needed partial derivatives, the (non extended) infinitesimal transformation of $f$ is given by

$$Uf=\xi_1\frac{\partial f}{\partial x_1}+\xi_2\frac{\partial f}{\partial x_2}+...+\xi_n\frac{\partial f}{\partial x_n}+\eta\frac{\partial f}{\partial y},$$ where

$$\xi_k=\left ( \frac{\partial \phi_k}{\partial a} \right )_{a_0}$$ $$\eta=\left ( \frac{\partial \psi}{\partial a} \right )_{a_0}$$ and $a_0$ is the value of the parameter which yields identity.

Now, how does one write the (non extended) $Uf$ when we have $m$ dependent variables $y_1,y_2,...,y_m$? Also, what would be a good text to read up more about it?

Answer 1

Found the answer at the FAS site, in the pdf file on page 33. ('Lie Group Invariant Finite Difference Schemes for the Neutron Difusion Equation', by Peter James Jaegers)

Given is the point transformation

$$x_k^{\prime}=\phi_k(x_1,x_2,...,x_n,y_1,y_2,...,y_m,a), \mbox{ } k=1...n$$ $$y_l^{\prime}=\psi_l(x_1,x_2,...,x_n,y_1,y_2,...,y_m,a), \mbox{ } l=1...m$$ where $a$ is the parameter of the group.

Then, if $f(x_1,x_2,...,x_n,y_1,y_2,...,y_m,a)$ is a smooth function and has the needed partial derivatives, the (non extended) infinitesimal transformation of $f$ is given by

$$Uf=\sum_{k=1}^n \xi_k\frac{\partial f}{\partial x_k}+\sum_{l=1}^m \eta_l\frac{\partial f}{\partial y_l},$$ where

$$\xi_k=\left ( \frac{\partial \phi_k}{\partial a} \right )_{a_0}$$ $$\eta_l=\left ( \frac{\partial \psi_l}{\partial a} \right )_{a_0}$$ and $a_0$ is the value of the parameter which yields identity.