Integral curves

Question

Suppose $f$ is a vector field. Let $g$ be an integral curve whose domain contains $[0,\infty)$. Now $\lim (g(t)) =p$ as $t \to \infty$. Then can we say that $f(p)$ is zero?

Answer 1

I am assuming that $f$ is continuous in all its variables, and each of the components of the integral curve are continuous. Let $g_i(t)$ be the $i$th component of the integral curve, and $f_i$ the $i$th component of $f$, and $p_i$ the $i$th component of $p$. Since $g_i(t) \to p_i$ we get that $|g_1(n+1)-g_1(n)| \to 0$ as $n \to \infty$. Using the Mean Value Theorem from Calculus we have a $n \leq c \leq n+1$ such that $$ |g_i(n+1) - g_i(n)| = |g_i'(c)| = |f_i(g_1(c),g_2(c),\ldots,g_m(c))|.$$ Taking limits as $n \to \infty$ (and thus as $c \to \infty$) on both sides and using continuity this implies that $$ \lim_{c \to \infty} |f_i(g_1(c),g_2(c),\ldots,g_m(c))|=0 \Rightarrow |f_i(p)|=0$$ since $\lim_{c\to \infty} (g_1(c),g_2(c),\ldots,g_m(c))=p$. So $f_i(p)=0$ for every $i$, and thus $f(p)=0$.