Derivatives of Vectors:   dV/dt|_A = dV/dt|_B + w_B/A X V

Hey we've been talking about derivatives of vectors all along - the velocity of a particle was the time derivative of the position of the particle, and the acceleration of a particle was the time derivative of the velocity of the particle, but we've never seen anything that even vaguely looked like the above formula.  We can find it in the text book as Equation 7/7 on page 547 but we still don't know what it means.  So what gives?
 
The above formula makes a mathematical statement of something we all know to be true.  It states that two people observing the same thing may very well see it differently!  To be more precise the above formula refers to two observers attached to two different coordinate systems.  Observer A is attached to and moving with coordinate system A.  Observer B is attached to and moving with coordinate system B.  Coordinate system B is rotating about some axis at some rate relative to system A.  This rotation is described by the angular velocity vector w_B/A in the above formula.  The two observers are both watching the same vector V.    Because the observers are rotating to one another they perceive different directions of the vector V.   That is one observer may see the vector V as being parallel to the X axis of their coordinate system, while the other observer may see the vector V as being parallel to the Y axis of their coordinate system (thus at this instant the X axis of one system is parallel to the Y axis of the other).  As the two observers perceive the vector V differently at different times, their calculations of the time derivative of the vector yield different results.  The above formula describes exactly the difference one could expect in the time derivatives calculated by the two observers.  Thus the formula makes a very powerful statement about the relationship between time derivatives of any vector measured by two different observers rotating relative to one another.

The general formula given above can be brought a little closer to reality by defining some of the elements involved more closely.  Coordinate system A, has axes X_A, Y_A, and Z_A.  Unit vectors in these directions shall be labeled I_A, J_A, and K_A.  Coordinate system B has axes X_B, Y_B, and Z_B.  Unit vectors in these directions shall be labeled I_B, J_B, and K_B.  As we discovered in our earlier investigation of spherical coordinates, unit vectors from one coordinate system can be related to another by a matrix equation.  The matrix relating the two sets of unit vectors is a 3 by 3 matrix known as a matrix of direction cosines.  This requires that we expand our notation conventions. We have distinguished vectors from scalars by putting the vectors in bold face.  We will distinguish 3 by 3 matrices from vectors and scalars by putting the matrices in underlined bold face.  Naming this rotation matrix R we know that the unit vectors can be related by an equation of the form (see the spherical coordinates page for some examples of rotation matrices):

I_B           |                |     I_A
J_B   =      |       R      |     J_A
K_B          |                |    K_A

Note that the matrix R changes with time as the coordinate systems rotate relative to one another.  Say the observer in system A measures the vector V to have components in system A that change with time as defined by the following:

V = VX_A(t) I_A + VY_A(t) J_A + VZ_A(t) K_A

As this observer perceives the I_A, J_A, and K_A vectors as fixed, the time derivative they would calculate would be expressed as:

dV/dt|_A  = d(VX_A(t))/dt I_A + d(VY_A(t))/dt J_A + d(VZ_A(t))/dt K_A

In developing the above expression, the unit vectors have been treated as constants.  The observer in system B sees things differently.  Based on the current value of the matrix of direction cosines, R, this observer sees a different set of components:

V = VX_B(t) I_B + VY_B(t) J_B + VZ_B(t) K_B

The component functions for observer B can be determined from those for observer A using the rotation matrix R.  When the observer in system B evaluates the time derivative of the vector, they perceive the unit vectors in system B to be a constant.  Thus:

dV/dt|_B  = d(VX_B(t))/dt I_B + d(VY_B(t))/dt J_B + d(VZ_B(t))/dt K_B

Even if we used our rotation matrix to convert one of these derivative vectors back into the other set of unit vectors, we still do not expect to them to be equal to one another.  This is because the observer in system A, thought their unit vectors were constants while the unit vectors in system B were rotating.  Conversely the observer in system B thought their unit vectors were constants and the unit vectors in system A were rotating.  Thus the two observers have a fundamental disagreement as to what is constant and what is changing.  Thus when they take derivatives, since they can't agree on what is constant, they can't possibly come to the same result.  Trouble!

This is where our formula comes in.  The formula states that the difference between the time derivatives calculated by the two observers is just equal to the angular velocity vector of one coordinate system relative to the other crossed into the vector that was being differentiated.  This is amazing in its simplicity considering the possible complexity of the rotation of one system relative to the other.  While all this may seem a bit esoteric, the implications are profound.  Newton's laws involved many vector time derivatives.  The velocity was the derivative of the position.  The acceleration was the time derivative of the velocity.  The sum of the external forces acting on a system is equal to the time derivative of the linear momentum of the system.  The sum of the external moments acting on a system is equal to the time derivative of the angular momentum of the system.  Thus if we are going to use Newton's laws we need to be able to differentiate vectors.  In doing so we can see great care needs to be taken to account for the effect of any rotation of the observer attempting to evaluate the time derivatives.  This leads to the concept of an inertial coordinate system.  But that's a story for another day.  For now we are happy with this revelation about how one's perspective influences one's observations!

So we have a deep philosophical statement of how one's perspective influences one's perception, so what?  How can we use this formula?  To see an example of the use of the above formula, we shall return to our consideration of spherical coordinates.  Click here to return to the mystical world of spherical coordinates!
 
Copyright (1998) by Nels Madsen.  Last Updated: February 28, 1998