Consider a one-parameter Lie group of transformations $\displaystyle \mathbf{x}^* = \mathbf{X}(\mathbf{x},\varepsilon)$. The infinitesimals are given by $\displaystyle \vec{\xi} = \left.\frac{\partial \mathbf{X}}{\partial \varepsilon}\right|_{\varepsilon=0}$ and the infinitesimal generator: $\displaystyle X = \vec{\xi}\cdot\nabla$. Let $\displaystyle \mathbf{x}^* = (x^*,y^*)$. Now what if we have $\displaystyle x^*=\frac{1}{\varepsilon}x$ and $\displaystyle y^* = \varepsilon y$? If we want to get the first component of $\displaystyle \vec{\xi}$, we need to differentiate $\displaystyle x^*$ with respect to $\displaystyle \varepsilon$ and evaluate it at $\displaystyle \varepsilon=0$ and we should divide by zero. How can it be solved?

Thanks,

Zoli