Fecko: Differential Geometry 2.3.1

[Rasalhague]

Show that the differential equations for finding an integral curve γ of a vector field
V have the form

$$\dot{x}^i(t)=V^i(x^1,...,x^n)$$.

According to the preceding paragraph,

$$x^i(t) \equiv x^i (\gamma (t)).$$

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

$$\dot{\gamma}(p) = \dot{\gamma} \circ \gamma (t):=\frac{\mathrm{d} }{\mathrm{d} t} x^i \circ \gamma(t) \bigg|_t [\gamma]_i = [\gamma],$$

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

$$\dot{x}^i(p)=(\dot{x^i \circ \gamma}) \circ \gamma (t)=\frac{\mathrm{d} }{\mathrm{d} t} x^i \circ x^i \circ \gamma(t) \bigg|_t [\gamma]_i = [\gamma]=\dot{\gamma}(p) \enspace ?$$

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?

 

[zhentil]

You're very much on the right track. $$\gamma (t)$$, in the local coordinates chosen, can be regarded as a vector valued function.

$$x_i(t) = x_i( \gamma(t) )$$

can be thought of as the i'th coordinate in this vector valued function.

However, you're wrong on the derivative part. $$\gamma ' (t)$$ in these coordinates is again a vector-valued function. If you want to be formal and consider $$x_i$$ as a function from R^n to R, then his notation doesn't make sense: $$x_i'(t)$$ is shorthand for $$\frac{d}{dt}x_i(\gamma(t))$$.

 

[Rasalhague]

The part I think I understand. I'm mainly going through all this so that we know we're talking the same language, as I wasn't always sure what you meant by a "vector-valued function." A function whose range is a subset of $\mathbb{R}^n$, n > 1? A function whose range is either $M$ or a subset of $\mathbb{R}^n$, n > 1? (Some authors, in this context, make "vector" synonymous with tangent vector, an element of $TM$, the tangent bundle of the particular manifold under discussion; but then, a vector might simply mean a vector with respect to any vector space.) Also when you say "q(w) is a function" do you mean this literally, or should I read it as "q is a function"?

Background, definitions of notation and terminology:

$M$ is a smooth manifold. $\gamma$ is a curve, that is, a function of the from

$$\gamma : (a,b) \rightarrow M, \enspace\enspace a,b \in \overline{\mathbb{R}}.$$

A tangent vector associated with point $P \in M$ is, according to the formalism Fecko is using in this section, an equivalence class of curves $[\gamma]$ such that for every pair of representatives $\gamma$ and $\sigma$, and every pair of charts $x$ and $y$, we have $\gamma (t_0) = \sigma (t_0) = P$, and

$$\frac{\mathrm{d} }{\mathrm{d} t} x \circ \gamma \bigg|_{t_0} = \frac{\mathrm{d} }{\mathrm{d} t} y \circ \sigma \bigg|_{t_0}.$$

EDIT: Oops, not for every pair of charts, just for one chart.

The coordinate presentation of a curve, $\gamma$, in a chart $x$ is the function

[tex]x \circ \gamma : (a,b) \rightarrow \mathbb{R}^n,[/itex]

letting $\circ$ mean the function which makes any necessary restrictions to the range of $\gamma$ before composing it with $x$, in case the range of the curve contains elements outside of the domain of $x$.

A vector field is a function

[tex]V:M \rightarrow TM,[/itex]

i.e. one whose inputs are points in $M$ and whose outputs are tangent vectors.


Now, to the part I'm working on... Hmm, I see now that the third equation in my post #1 can't be right, since $x^i : U \subseteq M \rightarrow \mathbb{R}$, so the composition $x^i \circ x^i$ has range the empty set, if $\circ$ is taken to mean "restrict and compose", or is meaningless, if $\circ$ is taken to mean simply "compose".

While I'm pondering what you wrote, it might be useful to quote Fecko in full.

Fecko: "An integral curve of a vector field V is then the curve γ on M, such that at each point of its image, the equivalence class [γ] given by the curve, coincides with the class V_P, given by the value of the field V in P. Put another way, from each point it reaches, it moves away exactly in the direction (as well as with the speed) dictated by the vector V_P . All this may be written as a succinct geometrical equation

$$\dot{\gamma} = V, \enspace\enspace\enspace \text{i.e } \dot{\gamma}(P) = V_P.$$

"(this is the equation for finding an integral curve γ of a vector field V in a “coordinate-free” form), where the symbol $\dot{\gamma}(P)$ denotes the tangent vector to the curve γ at the point P (i.e. the equivalence class [γ], given by the curve γ at the point P). If the vectors on both sides of this equation are decomposed with respect to a coordinate basis, a system of differential equations for the functions xⁱ(t) ≡ xⁱ(γ(t)) (for the coordinate presentation of the curve to be found) is obtained."

 

[Rasalhague]

So, if we let $(V,x):\mathbb{R}^n \rightarrow \mathbb{R}^n$ denote the coordinate presentation of a particular, given vector field V, then the differential equation is just setting the following condition on an unspecified function $x \circ \gamma : I \subseteq \mathbb{R} \rightarrow \mathbb{R}^n$?

$$\frac{\mathrm{d} }{\mathrm{d} t} x \circ \gamma(t) \bigg|_{t_0} = (x,V)(a)$$

where $I=(a,b)$, an open interval with endpoints $a$ and $b$. Or do we also need to set some extra condition relating V to gamma, namely gamma'(P) = V_P, and some rule relating V to its coordinate presentation?

EDIT: I accidentally switched the order of x and V in (V,x). It's not significant.

 

[Rasalhague]

Any idea what this means?

Fecko: "From the definition of linear combination in (2.2.2) one can see that all vectors at the point P share the same values of $x^i(P)$ and the property by which they can be distinguished from one another is by the values of the coefficients $a^i=\dot{x}^i(0)$."

The xⁱ takes as its input a point in M, yet is described as if it's a property of a particular vector. Is it just saying all tangent vectors at P are at P? The dotted version is a function of real numbers, unlike the dotted gamma in 2.3. But, while trying to decipher the dotted x, I see now that there are actually two contradictory definitions of dotted gamma in play; in 2.2.2, $\dot{\gamma}$ is defined as $[ \gamma ]$, a tangent vector seen as a certain equivalence class of curves. But later, in 2.3, $\dot{\gamma}$ is defined as V, which earlier stood for a vector field. Its inputs are points of M, in contrast to your gamma prime, whose inputs are real numbers. I'm guessing the change he makes is because earlier only one point was being considered, so there was no need to distinguish between tangent vectors at different points. But, as soon as the curve as a whole is considered, by the first definition, dotted gamma would mean something different, a different equivalance class of curves, at each point, so it would have infinitely many referents.