Page 331 - Introduction to Continuum Mechanics

Change of Frame 315

The distance between two material points is called a frame-indifferent (or objective) scalar
because it is the same for any two observers. On the other hand, the speed of a material point
obviously depends on the observers as the observers in general move relative to each other.
The speed is therefore not frame indifferent (non-objective). We see therefore, that while a
scalar is by definition coordinate-invariant, it is not necessarily frame-indifferent (or frame-
invariant).
The position vector and the velocity vector of a material point are obviously dependent on
the observer. They are examples of vectors that are not frame indifferent. On the other hand,
the vector connecting two material points, and the relative velocity of two material points are
examples of frame indifferent vectors.
Let the position vector of two material points be \j, \2 m tne unstarred frame and x|, x|
in the starred frame, then we have from Eq. (5.32.la)

Thus,

or,

where b and b* denote the same vector connecting the two material points.
Let T be a tensor which transforms a frame-indifferent vector b into a frame-indifferent
vector e, i.e.,

let T * be the same tensor as observed by the starred- frame, then

Now since c* = Qc, b* = Qb, therefore,

i.e.,

Thus,

Summarizing the above, we define that, in a change of frame,