- #1

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Let ##\gamma:\mathbb{R}\supset I\to M## be a curve that we'll parameterize using ##t##, i.e. ##\gamma(t)\in M##. It's stated that:

Immediately after there's an example: if ##X=x\partial_x+y\partial_y##, then ##dx/dt=x## and ##dy/dt=y##, which gives the integral curve passing through ##(a,b)## at ##t=0## as ##\gamma(t)=(ae^t,be^t)##.If ##\gamma(t)## has coordinates ##x^i(t)## and [a vector field] ##X## has components ##X^i##, finding the integral curve associated with ##X## reduces to solving a set of coupled first-order differential equations, $$\frac{d}{dt}x^i=X^i(x^1(t),\ldots,x^n(t))$$

- Now from the context, provided we're talking about only one curve ##\gamma##, shouldn't ##X## actually be the
**restriction of the vector field to the curve**##\gamma##, rather than the vector field itself? - Referring to the phrase "If ##\gamma(t)## has coordinates ##x^i(t)##...", I'm guessing it's unlikely that
**all**the points on the curve belong to a single chart. So how can we claim only one coordinate system ##x^i## to represent the coordinates of**all**the points on the curve? Won't we have to adjust the coordinates according to the chart?

e.g. if some ##p,p'\in\gamma(I)## are covered by different charts, and if the coordinates of ##p## are ##x^i##, won't the coordinates of ##p'## have to be characterized by an entirely different coordinate system (e.g. some ##y^i##)?