Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am studying Homegeneity lemma. I am not understanding the following paragraph:

Given any fixed unit vector $c \in S^n$, consider the differential equations

$\frac{dx_i}{dt} = c f(x_1,x_2,\ldots,x_n)$ for $i=1,2,\ldots,n$, where $f$ is a smooth function from $\mathbb{R}^n$ to $\mathbb{R}$ with $f(x)=0$ for outside the unit sphere and on sphere and $f(x)>0$ for inside the unit interval.

For any $y \in \mathbb{R}^n$ these equations have a unique solution $x = x(t)$, defined all real numbers which satisfies the initial condition $x(0)=y$. We will use the notation $x ( t ) = F_t (y)$ for this solution. Then clearly

  1. $F_t(y)$ is defined for all $t$ and $y$ and depends smoothly on $t$ and $y$,
  2. $F_0(y) =y$,
  3. $F_(t+s)(y) = F_t \circ F_s(y)$.
share|improve this question

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.