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Topic: Lovelock and Rund: Star shaped set of points on a manifold
Replies: 7   Last Post: Jan 17, 2013 2:39 PM

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 quasi Posts: 12,067 Registered: 7/15/05
Re: Lovelock and Rund: Star shaped set of points on a manifold
Posted: Jan 16, 2013 3:34 PM

Hetware wrote:
>This is from Lovelock and Rund, _Tensors, Differential Forms,
>and Variational Principles_, pg. 142.
>
>"[f]or a given point P on X_n let us choose our coordinates
>such that x^1=...=x^n=0 at P, after which we construct an
>open set U on X_n which is defined by the property that for
>any point Q element of U with coordinates x^h, the segment
>consisting of the points with coordinates tx^h, 0<=t<=1, is
>also contained in U."
>
>I'm having a bit of trouble grasping that concept. Let's
>take R^2, for example. I choose a point in the middle of my
>paper, and call it {0,0}. I now declare it to be a member of
>some open set U.

You mean, let U be an open set containing (0,0).

>The smallest possible U is some infinitesimal open disk
>centered on P.

There's no smallest possible U.

Perhaps you meant to say something like:

U can be chosen as an open disk centered at (0,0) with an
arbitrarily chosen positive radius, regardless of how small.

>If I chose some point in U (whatever that means)

In this context it means that the property to be discussed
holds for all points of U.

>and multiply it by some t from the closed interval [0, 1], I
>get an infinitesimal curve segment including P and Q.

Since you are in R^2, you get the line segment from P to Q,
that is, the line segment from (0,0) to Q.

The length of the segment depends on the choice of Q, but it's
an actual line segment (degenerate if Q = (0,0).

>So am I correct in understanding that the "star shaped" set U
>is the infinitesimal disk centered on P, with parametrized

No.

U is star-shaped in the sense that any point in U is "visible"
from (0,0).

Thus, U need not be a disk.

For example, any convex open set containing (0,0) qualifies.

But U need not be convex either.

For example, the interior of a Jewish star centered at (0,0)
also qualifies.

Staying with the religious theme, the interior of a Christian
cross centered at (0,0) qualifies as well.

Perhaps now you get the picture.

>If I follow a constructive approach, it seems that U cannot
>have a finite extension.

I have no idea what you mean by that.

>If U were chosen such that some x^h > 1 at some point Q in U,
>the whole thing would explode, since both my coordinate
>magnitude and my parameter could be greater than 1, thus
>identifying points with yet larger coordinate magnitudes.

No, the parameter t is required to satisfy 0 <= t <= 1, so
the points t*x^h stay on the line segment from (0,0) to Q.

>Is any ball with x^i < 1 a permissible choice for U?

An open ball of radius 1 centered at (0,0) is a legal choice
for U, but the concept of

"an open star-shaped region containing (0,0)"

allows for lots of other possible shapes

quasi

Date Subject Author
1/16/13 quasi
1/16/13 quasi
1/16/13 Butch Malahide
1/16/13 quasi
1/16/13 William Elliot
1/17/13 Shmuel (Seymour J.) Metz