Integral Curves Sample Clauses

Integral Curves. A vector field gives a direction and rate for every point on a mani- fold. We can start at any point and go in the direction specified by the vector field, tracing out a parametric curve on the manifold. This curve is an integral curve of the vector field. More formally, let v be a vector field on the manifold M. An integral curve γv : R → M of v is a parametric path on M satisfying D(f ◦ γv )(t) = v(f)(γv (t)) = (v(f) ◦ γv )(t) (3.24) m m m m γv (0) = m, (3.25) for arbitrary functions f on the manifold, with real values or struc- tured real values. The rate of change of a function along an inte- gral curve is the vector field applied to the function evaluated at the appropriate place along the curve. Often we will simply write γ, rather than γv . Another useful variation is φv(m) = γv (t). m t m We can recover the differential equations satisfied by a coor- dinate representation of the integral curve by letting f = χ, the coordinate function, and letting σ = χ ◦ γ be the coordinate path corresponding to the curve γ. Then the derivative of the coordi- nate path σ is Dσ(t) = D(χ ◦ γ)(t) = (v(χ) ◦ γ)(t) = (v(χ) ◦ χ−1 ◦ χ ◦ γ)(t) = (b ◦ σ)(t), (3.26)