Editing VE (section)

====Standard Presentation [the Virial of Clausius (1870)]====
The trace of the tensor virial equation (TVE), which is obtained by identifying the trace of each term in the TVE, produces the scalar virial equation, which is widely referenced and used by the astrophysics community.  More specifically, setting, 
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Description
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&nbsp;
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[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]
Reference
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<math>I = \sum\limits_{i=1,3} I_{ii}</math>
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<math>=</math>
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<math>\int\limits_V \rho (\vec{x}) |\vec{x}|^2 d^3x  </math>
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=
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scalar moment of inertia
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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[Eqs. (3) &amp; (5), p. 16]
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<math>T_\mathrm{kin} = \sum\limits_{i=1,3} \mathfrak{T}_{ii}</math>
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<math>=</math>
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<math>\frac{1}{2} \int\limits_V \rho |\vec{v}|^2  d^3x </math>
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=
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total (ordered) kinetic energy
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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[Eq. (8), p. 16]
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<math>W_\mathrm{grav} = \sum\limits_{i=1,3} \mathfrak{W}_{ii}</math>
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<math>\equiv</math>
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<math>- \int\limits_V \rho x_i \frac{\partial \Phi}{\partial x_i} d^3x </math>
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=
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gravitational potential energy
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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[Eq. (18), p. 18]
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<math>S_\mathrm{therm} = \frac{1}{2} \sum\limits_{i=1,3} \delta_{ii}\Pi</math>
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<math>=</math>
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<math>\frac{3}{2} \int\limits_V P d^3x </math>
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=
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total thermal (random kinetic) energy
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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[Eq. (7), p. 16]
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the scalar virial equation is,
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<math>\frac{1}{2} \frac{d^2 I}{dt^2}</math>
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<math>=</math>
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<math>2 (T_\mathrm{kin} + S_\mathrm{therm}) +  W_\mathrm{grav} \, ;</math>
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and, for a stationary state, we have the equilibrium condition that is broadly referred to as the,
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<span id="TVE"><font color="#770000">'''Scalar Virial Theorm'''</font></span><br />
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<math>2 (T_\mathrm{kin} + S_\mathrm{therm}) +  W_\mathrm{grav} </math>
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<math>=</math>
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<math>0 \, .</math>
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[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 213, Eq. (4-79)
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(In a footnote to their Equation 4-79, [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>] point out that the ''scalar virial theorem'' was first proved by R. Clausius in 1870; see various links to this work under our [[VE#Related_Discussions|"Related Discussions" subsection, below]].)