I'd like to define a weak vector field in $\Bbb R^2$ that is tangential to a family of sine curves at each point. I define the family of sine curves as: $A_n\sin(x)$; $n=1,2,3,...$ and $A_n$ is a parameter; $A_n\in \Bbb Q^2.$ The vector field is weak in $\Bbb R^2$ but is complete in $\Bbb Q^2.$ I'm have trouble with finding the correct rational $A_n$ and expressing this vector field.
Edit: Viewing the family of sine curves as the solution curves to the differential equation: $y''+y=0$ with the initial condition as: $y(0)=0$ in $\Bbb R^2$ is helpful and seems help with the problem in $\Bbb R^2$. Not sure what to do about $\Bbb Q^2$.
I took Vector Calculus two years ago so I'm a little rusty. I know intuitively what to do but am struggling to capture this intuition in concrete notation.
A weak vector field is basically a vector field with incomplete information. So there is not a vector at every single point in the vector field. Researchers study weak vector fields because they are easier to work with analytically.
$1)$ Please be descriptive and rigorous in defining the vector field.
$2)$ Once one rigorously defines this vector field in $\Bbb Q^2,$ how does one take a kind of "limit" so as to make this vector field complete in $\Bbb R^2?$