I have seen authors like John Lee in his "Smooth Manifolds" book, use these two terms as being different, but without , AFAIK, explaining the difference between the two.
I assume it may be that Euclidean n-space is the manifold R^n with a preferred , or "default" chart, but I am not sure of this.
> I have seen authors like John Lee in his "Smooth Manifolds" book, use these two terms as being different, but without , AFAIK, explaining the difference between the two.
> I assume it may be that Euclidean n-space is the manifold R^n with a preferred , or "default" chart, but I am not sure of this.
> I have seen authors like John Lee in his "Smooth Manifolds" book, use > these two terms as being different, but without , AFAIK, explaining the > difference between the two.
> I assume it may be that Euclidean n-space is the manifold R^n with a > preferred , or "default" chart, but I am not sure of this.
> Anyone Else Know.?.
> Thanks In Advance.
I would probably say that R^n is Euclidean n-space together with a coodinate system. Euclidean n-space has no preferred point "the origin" nor preferred directions for the coordinate planes, while R^n has all of those.
> > I have seen authors like John Lee in his "Smooth > Manifolds" book, use > > these two terms as being different, but without , > AFAIK, explaining the > > difference between the two.
> > I assume it may be that Euclidean n-space is the > manifold R^n with a > > preferred , or "default" chart, but I am not sure > of this.
> > Anyone Else Know.?.
> > Thanks In Advance.
> I would probably say that R^n is Euclidean n-space > together with a > coodinate system. Euclidean n-space has no preferred > point "the > origin" nor preferred directions for the coordinate > planes, while R^n > has all of those.
> > > I have seen authors like John Lee in his > "Smooth > > Manifolds" book, use > > > these two terms as being different, but without > , > > AFAIK, explaining the > > > difference between the two.
> > > I assume it may be that Euclidean n-space is > the > > manifold R^n with a > > > preferred , or "default" chart, but I am not > sure > > of this.
> > > Anyone Else Know.?.
> > > Thanks In Advance.
> > I would probably say that R^n is Euclidean n-space > > together with a > > coodinate system. Euclidean n-space has no > preferred > > point "the > > origin" nor preferred directions for the > coordinate > > planes, while R^n > > has all of those.
> > > > I have seen authors like John Lee in his > > "Smooth > > > Manifolds" book, use > > > > these two terms as being different, but without > > , > > > AFAIK, explaining the > > > > difference between the two.
> > > > I assume it may be that Euclidean n-space is > > the > > > manifold R^n with a > > > > preferred , or "default" chart, but I am not > > sure > > > of this.
> > > > Anyone Else Know.?.
> > > > Thanks In Advance.
> > > I would probably say that R^n is Euclidean n-space > > > together with a > > > coodinate system. Euclidean n-space has no > > preferred > > > point "the > > > origin" nor preferred directions for the > > coordinate > > > planes, while R^n > > > has all of those.
> So is Euclidean n-space what's referred to > by "affine" space? Thanks, Rich Peterson
An affine space does not have a metric structure (yet). One can play around with setting up axioms for the metric, without specifying the origin or axes. A "preferred" metric satisfying the Parallelogram Law will do. (It is a classical exercise how to obtain the inner product in the underlying vector space out of it by polarization -- then you can measure angles etc.) But under merely affine transformations, angles get distorted.
> An affine space does not have a metric structure (yet). One can play > around with setting up axioms for the metric, without specifying the > origin or axes. A "preferred" metric satisfying the Parallelogram Law will > do. (It is a classical exercise how to obtain the inner product in the > underlying vector space out of it by polarization -- then you can measure > angles etc.) But under merely affine transformations, angles get > distorted.
> Cheers, ZVK(Slavek).
Agreed. Euclidean space has congruence, but affine space doesn't.
> > I have seen authors like John Lee in his "Smooth > Manifolds" book, use these two terms as being > different, but without , AFAIK, explaining the > difference between the two.
> > I assume it may be that Euclidean n-space is the > manifold R^n with a preferred , or "default" chart, > but I am not sure of this.
> > Anyone Else Know.?.
> > Thanks In Advance.
ok , mr ' guest ' , we are on to you.
we know that usually when ' guest ' replies with complete nonsense its probably -> MUSATOV.
dear musatov , God knows about your sins ; your insults under the name of ' guest '.
unless masatov believes in a god more retarded than himself.
but that is of course not possible , the creator of the universe - if he even exists - could not possibly be more retarded than musatov , otherwise he would not be able to create.
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