This is actually a very old question that now I have to face it again and look for answer of it. Suppose $f:\mathbb{R}^n\to \mathbb{R}, f(x_1,x_2,...,x_n)=y$ is a function. What is the difference between:
- $\frac{\partial f}{\partial x_i}$
- $\frac{df}{dx_i}$ if this means anything at all?
I am reading this book and the following passage is part of the book:
...it would be appropriate to introduce a scaled time $\tau$ via
$$\tau=\epsilon^2 t$$
and regard $u$ as depending both on $t$ and $\tau$, and having no explicit dependence on $\epsilon$; $t$ and $\tau$ will be treated as mutually independent. Correspondingly, the time differentiation should be transformed as
$$\frac{d}{dt}\to\frac{\partial}{\partial t}+\epsilon^2 \frac{\partial}{\partial \tau} $$
Where $u=X-X_0$ and $X_0$ is a stable answer for following differential equation: $$\frac{dX}{dt}=F(X)$$