[Insight-developers] Inner Product != Scalar Product

[Luis Ibanez]

This is a multi-part message in MIME format.
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Hi,

To better explain this we need some diagrams, 
so I prepared a PDF file with some description
of the differences between the inner product and
the scalar product, the role of the Metric Tensor
and its relationship with covariant and contravariant
components.

All our conflicts are originated in our need for
affine transformations (or more complex transforms).
As soon as we abandon the orthogonal isometric 
coordinate system, things are less trivial and some 
reflexes have to be reanalized, like for example 
using the inner product for computing the norm of 
a vector.

The attached PDF file makes 53Kb, I'm cc'ing the
list but if the file is to big I'll better check 
it in InsightDocuments.



Luis



-----------------

George Stetten wrote:
> 
> Hi Luis,
> 
> What's the purpose?  to permit inner products to incorporate a different coordinate system than that
> which the vectors are defined.... right?   Andrew Thall (he's still there at CMU) was an ace at the
> metric tensor, he TA'd Steve Pizers course that I took in differential geometry, in case we need
> backup.  I think I might be able to serve here as before, in that if I can understand it everyone
> will....
> 
> So, why do we need this metric tensor function?
> 
> George
> 
> "Miller, James V (CRD)" wrote:
> 
> > Luis,
> >
> > Should the vector classes have another "inner product" method wrt to a specified metric tensor?
> > Somewhere deep in the recesses of math knowledge I seem to recall using a form like
> >
> >         <Mv, w> = v * M * w'
> >
> > Should we have methods in vector and covariant vector that perform the above calculation?
> >
> > Jim
> >
> >
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