- #1
Rasalhague
- 1,387
- 2
Show that the differential equations for finding an integral curve γ of a vector field
V have the form
[tex]\dot{x}^i(t)=V^i(x^1,...,x^n)[/tex].
According to the preceding paragraph,
[tex]x^i(t) \equiv x^i (\gamma (t)).[/tex]
For now, I have the lowly ambition of trying to understand the notation. I think the xi on the left of the "quoted" equation is a coordinate presentation of a curve (Fecko's curves are functions from an interval to a smooth manifold), here denoted by the symbol which in other contexts may mean one of the coordinate/component functions of a chart, or a representative value of such a function. By contrast, I think the x1 etc. on the right are real numbers. Is this the likely meaning?
If I managed to follow the preceding discussion, the dot means
[tex]\dot{\gamma}(p) = \dot{\gamma} \circ \gamma (t):=\frac{\mathrm{d} }{\mathrm{d} t} x^i \circ \gamma(t) \bigg|_t [\gamma]_i = [\gamma],[/tex]
where [ ] denotes the equivalence class of curves which is one way to formalise the concept of a tangent vector, and the indices are summed over, and [γ]i is a basis vector, so
[tex]\dot{x}^i(p)=(\dot{x^i \circ \gamma}) \circ \gamma (t)=\frac{\mathrm{d} }{\mathrm{d} t} x^i \circ x^i \circ \gamma(t) \bigg|_t [\gamma]_i = [\gamma]=\dot{\gamma}(p) \enspace ?[/tex]
So the left side of the quoted equation is just a way of denoting the coordinate presentation of a tangent vector, defined as a certain equivalence class of curves, namely by denoting an integral curve belonging to this class by the name of the coordinate system itself?
Last edited: