Hello All
I am trying to read the text book Tensor analysis by IS Sokolnikoff and stuck in
chaper 1, there is some confusion in my mind regarding orthogonal transformations,
change of basis and and a transformation representing deformation of space.
Please bear with me if my confusions are unfounded.
1) Is is safe to say that transformation representing deformation of space acts on
vectors where as transformations representing the change of basis acts on
components?. (if yes then the following questions are resolved and need not be
looked)
2) Sokolnikoff builds up the reasoning for orthogonal transformations based on
deformation of space (i.e orthogonal transofrmations keep the length of vector
constant), although at the end it appears that he defines it in terms of determinant
of a general transformation matrix working on components being equal to one . If
orthogonal transformations are defined in terms of length of vector being constant
and if transormation of basis acts on vectors then doesnt it mean any transformation
representing a change of basis is orthogonal. (which is of course not true) ( if
orthogonal transormations are defined in terms of determinant of transformation
matrix then we can have both orthogonal transformations representing deformation of
space and orthogonal transformation representing the change of basis.)
3) When discussing diagonal forms there are instances when the transformation
representing the change of basis are deemed orthogonal so it appears that the
orthogonal nature is defined with respect to the determinant of matrix and not with
respect to length of vector
Sorry for the random nature of questions on a trivial topic for many here.
Thanks in advance
nk