16,1.2 Rate of Change of Vectors in Rotating Frames

We have resolved vectors into components along coordinate axes in several coordinate systems. We shall now derive the relationship between the time derivatives of an arbitrary vector resolved along the coordinate axes of one system and the derivatives of the components in a different system. For definiteness, we consider the geomagnetic field vector in the body system and reference system, previously introduced. We wish to compare the time derivatives of the field measured by magnetometers fixed in* a rotating spacecraft with the derivatives measured by a (possibly fictitious) set of magnetometers traveling with the spacecraft but with a fixed orientation relative to the reference frame. If we denote the components of the vector in the reference system by a' =(at,a2,a3)T and the components in the body by a=(au,ac,a„)T, then, according to Eq. (12-4), a=i4a' (16-16)

The time variation of the components of a is due to the time variation of both A and a'- The former represents the variation due to the change of the relative orientation of the two reference systems. The product rule for differentiation gives

The first term on the right can be written, using Eq. (16-7), as

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