UNIT VECTOR DIFFERENTIATION IN SPHERICAL COORDINATES

Consider the following spherical coordinate system with unit vectors  (note that  is an intermediate unit vector orthagonal to ):

 

 

From the definition of differential we can write

Physical meaning of each term:

    : Change in  when only  is varied (r and  are held fixed)

    :Change in  when only  is varied (r and  are held fixed)

    :Change in  when only r is varied ( and  are held fixed)

 

To find , we distrub the system by :

 

 

Select  (equal to unit vector). Then,

(Note that we can also get )

And we have

Finally, if we divide both sides by dt, we get the time derivative of unit vector :