Showing that v(f)=0 Implies a Function is Constant on a Torus for v = $\partial_1 +\lambda \partial_2$

Question

On a torus we are given the vector field: $ v=\partial_1 +\lambda \partial_2$, and asked to show that if $\lambda$ is irrational then $v(f)=0$ implies that f is constant. I know that if $\lambda$ is irrational then the integral curves of $v$ densely cover the torus and if f is constant on them then it has to be constant on the whole torus. But I don't see how to proceed to show that it's constant. Any help is appreciated.

Answer 1

Recall that if $t$ is the parameter along the integral curves of $v$, so that

$v = \dfrac{\partial}{\partial t}, \tag 1$

then

$v(f) = \dfrac{\partial f}{\partial t}; \tag 2$

thus if

$v(f) = 0, \tag 3$

then

$\dfrac{\partial f}{\partial t} = 0, \tag 4$

showing $f$ is constant along these trajectories.