Dr/Dt world coordinates problem

[yetar]

Homework Statement

In rigid body, r is the position of a rigid body relative to world coordinates,
dr/dt is the rate of change of r in world coordinates.
I encountered the term Dr/Dt as the rate of change of r in world coordinates relative to the body coordinates system.
And that Dr/Dt = dr/dt-wxr
My question is, how does Dr/Dt is called? (terminology)
Also the operator D/Dt has the same rules as the operator d/dt, how do you proove it? There is some relationship to proove this, but what is this relation ship?

Homework Equations

Dr/Dt = dr/dt-wxr

The Attempt at a Solution

I tried to use Dr/Dt = dr/dt-wxr to proove that both operators has the same rules, but didnt succed.
In matter of fact, I found some contradiction.
For instance: d(AxB)/dt = Ax(dB/dt)+(dA/dt)xB
So D(AxB)/Dt = Ax(DB/Dt)+(DA/Dt)xB should also be true.
However using Dr/Dt = dr/dt-wxr showed that there is inequality.

Thank you.

 

[christianjb]

I prefer the notation

dr^s/dt=dr^b/dt-wxr

where r^s = space fixed r (with subscript s) and r^b = body fixed r

The d/dt operator is the same in both cases.

There's a good, but lengthy article here

http://python.rice.edu/~chem630/Polyatomics.pdf