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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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Posts: 12,067
Registered: 7/15/05
Re: Lovelock and Rund: Star shaped set of points on a manifold
Posted: Jan 16, 2013 11:40 PM
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Hetware wrote:
>Now I have another question along the same lines.
>On page 157

[Lovelock and Rund, _Tensors, Differential Forms
and Variational Principles_]

>is the following:
>"Let G be a (p+1)-dimensional region of X_n which is
>star-shaped relative to some point O in the interior of G and
>bounded by a closed p-dimensional subspace @G. It is assumed
>that the latter is covered by a finite number of coordinate
>neighborhoods U_1, U_2, ..., such that the corresponding
> parametrizations are of class C^1. If u^1,...,u^p denote
>the set of parameters corresponding to U_1, a region B of @G is
>determined by a closed set b in the domain R_p of the variables
>u^1,...,u^p. The region B is represented parametrically by the equations
>E^j = E^j(u^a), u^a element of b (a = 1,...,p)
>where E^j refer to a special coordinate system of X_n whose
>origin is located at O. By hypothesis, the set of points
>C(B) = {tE^j(u^a): 0<=t<=1, u^a element of b}
>defines a (p+1)-dimensional region of X_n which is contained
>entirely in G."
>According to the definition of a star-shaped region, it seems
>that G should be an open region.

No, a star-shaped region need not be open.

>I believe that means that G and @G are disjoint. That is, they
>share no points in common.

No, unless by "region". the author means "open region", G need
not be open. Thus, G and @G need not be disjoint.

However, presumably the author intends that @G is not just the
boundary of G, but also of the boundary of the interior of G.

>The specified coordinates E^j = E^j(u^a) represent points
>in @G. Since t is defined on [0,1] (inclusive), the
>E^j(u^a) are elements of C(B); but, since @G and G are

But as I mentioned above, G and @G may or may not be disjoint.
Still, they _might_ be disjoint, so for the sake of argument,
let's assume they are.

>"C(B) = {tE^j(u^a): 0<=t<=1, u^a element of b} defines a
>(p+1)-dimensional region of X_n which is contained entirely
>in G,"
>seem incorrect.
>Is the book wrong, or am I wrong?

Yes, it does appear to be an error.

There are two possible corrections ...

(1) In the definition of C(B), change

0<=t<=1 to 0<t<1.


(2) Change the conclusion from

... defines a (p+1)-dimensional region of X_n which is
contained entirely in the G.


... defines a (p+1)-dimensional region of X_n which is
contained entirely in the G U @G

I suspect that with one of the two corrections I suggested above
(it may not matter which), the arguments in the book which depend
on that excerpt will go through without any real problems, so all
in all, a minor error with no real impact.


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