Radial and Radial-Normal Unit Vector Differentials in D₁.

In this Appendix, proofs of the differentials of unit vectors in D₁ as represented by (2.7) and (2.8) are given.

Consider the vector in D₁.

(F1)

Differentiating this with respect to the time in D₁.

(F2)

Assume now that there is only radial normal motion, i.e., then

(F3)

Because this motion is only a radial normal one, the right hand side can be equated to a simple velocity term thus

(F4)

This must be valid for all values of w including boundary conditions. The lower condition is trivial, (when w = 0, u = 0), but at the upper condition of Terminal Spatial Velocity in the radial normal direction, i.e. w s = c, the left hand side of (F4) becomes

(F5)

At this boundary, temporal velocity is zero and spatial velocity is equal to the magnitude of Existence Velocity and therefore the right hand side of (F4) can be written

(F6)

Thus from (F5) and (F6)

(F7)

A similar proof exists for

(F8)

These relationships exist because the Spatial Terminal Velocity in the radial normal direction is different from the magnitude of Existence Velocity in this Domain.

On to the next Section: Appendix G

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