Dear Jesper:
Thanks! It's much more clear now.
Hedok
Jesper Andersson wrote:
> Dear Hedok,
>
>
>>
>> Thank you for your response. After thinking about your example and
>> applying your example to 3D, instead of 1D, I have one more question.
>> I must be naive in the way I'm thinking about Jacobian determinant so
>> please correct me if I'm wrong about this.
>> If I think of the determinant like a 3x3 tensor, and orthogonalize
>> it, I end up 3 eigenvalues. The product of the three eigenvalues is
>> the amount the volume being shrunk(<1) or strethched(>1). The
>> question is that the sign gets lost if two of the eigenvalues are
>> negatives, and the negative determinant only occurs if one, or all,
>> of the eigenvalues is negative.
>
> It does get a bit trickier when one tries to think about it in 3D.
>
> Maybe it is better to think about it in terms of handness, as in
> "right handed" or "left handed" coordinate systems. If you stick out
> your right-hand long and index finger and your thumb to make a right
> handed coordinate system in front of you, and define the directions
> that the fingers point in as positive. Now rotate your lower arm so
> your thumb points to the floor instead. You'll se that you thumb and
> your long finger are now pointing in the previously negative
> directions, i.e. you have two sign changes, i.e. the Jacobian is still
> positive. But that is what you want, because a rotation is certainly
> an "allowed" transform.
>
> Good luck Jesper
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