quasi
Posts:
12,018
Registered:
7/15/05


Re: Lovelock and Rund: Star shaped set of points on a manifold
Posted:
Jan 16, 2013 11:40 PM


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 >starshaped relative to some point O in the interior of G and >bounded by a closed pdimensional 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 starshaped region, it seems >that G should be an open region.
No, a starshaped 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 >disjoint,
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.
Or
(2) Change the conclusion from
... defines a (p+1)dimensional region of X_n which is contained entirely in the G.
to
... 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.
quasi

