TITLE: Rotation from Goldstein's Classical Mechanics
QUESTION [3 upvotes]: I apologize for the ambiguity in my title.
It was rather difficult to figure out what is the most appropriate title for my questions.
My questions come from chapter 4 and chapter 5 of Goldstein, which are both about rotations.
First regards to the paragraph following equation (4.84), which states
$dG_i = a_{ji}dG_j' +da_{ji}G_j'$ where prime coordinates are the body coordinate, and unprimed represents the space coordinate. $a$ represents the transformation matrix from space to body.
The paragraph states

It is no loss of generality to take the space and body axes as
  instantaneously coincident at the time t. Components in the two
  systems will then be the same instantaneously , but differentials will
  not be the same, since the two systems are revolving relative to each
  other. Thus, $G'_j=G_j$ but $a_{ji}dG_j' = dG_i'$.

I get its reasoning, except the conclusion. I agree that the differentials are different respect to the two coordinate system, but why does that imply $a_{ji}dG_j' = dG_i'$?
Second question I have comes from chapter 5, where it is stated (in section 1),

Any difference in the angular velocity vectors at two arbitrary
  points must lie along the line joining the two points.

Why is that true?

REPLY [3 votes]: Answer to first question:
At the instant being considered, the space and body axes are identical, so at that moment the matrix $a$ that relates the two sets of axes is simply the identity matrix.  $dG'$ is a vector, so $a_{ji}dG_j' = dG_i'$ is simply equivalent to the statement that with $I$ the identity matrix and $V$ an arbitrary vector, $IV=V$.
Answer to second question:
Immediately prior to the statement in question, the book has just derived that $(\omega_1 - \omega_2 ) \times R = 0$.  Using one possible definition for the cross product, that equation is equivalent to $\left\| (\omega_1 - \omega_2 ) \right\| \left\| R \right\| \sin \theta \ n = 0$, where $n$ is a unit vector perpendicular to the plane containing $(\omega_1 - \omega_2 ) $ and $R$, and $\theta$ is the angle between $(\omega_1 - \omega_2 ) $ and $R$.  $n$ is not the null vector, and $R$ is presumed to not be the null vector, so if $(\omega_1 - \omega_2 ) $ is also not the null vector, the only other possible way for that equation to hold is if $\sin \theta = 0$, i.e. $\theta = 0$.  I.e. , $(\omega_1 - \omega_2 )$ must be parallel to $R$, which is what the book means by the sentence in question.
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