Meaning of isomorphism/diffeomorphism ## f: R^n\to M^m##

[mathwonk]

you are probably right but just as an intuitive response, what about a vector field that equals zero at every point of the x axis? then the unique solution through a point of the x-axis would be the constant ciurve at that point. I.e, no solution could flow along the x-axis since the velocity is always zero there.

 

[zinq]

I meant — by a vector field tangent to all the curves G_c — a non-zero vector field tangent to these curves at each point. (One way to define such a field uniquely is to choose the vector of length = 1 having positive x-component that is tangent

"The" ordinary differential equation defined by the family of curves (for any c ∈ ℝ) G_c = {(x, (x-c)³) | x ∈ ℝ} would be any non-zero vector field V(x,y) tangent at every point (a,b) to the unique curve G_c that passes through (a,b). (This would be the curve G_c for the unique number c that satisfies (a-c)³ = b.)

But this vector field would be nowhere zero, so would not have zero velocity anywhere.