Editing VE (section)

====Derivation====
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>, &sect;11(b), p. 22] The second-order (tensor) virial equations <font color="#007700">are obtained by multiplying [the Euler equation] by <math>x_j</math> and integrating over the volume, <math>V</math></font>.  Specifically, again using our H_Book variable notation,
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<math>\int\limits_V \rho \frac{dv_i}{dt} x_j d^3x</math>
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<math>=</math>
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<math>- \int\limits_V x_j \frac{\partial P}{\partial x_i} d^3x - \int\limits_V \rho x_j \frac{\partial \Phi}{\partial x_i} d^3x \, ,</math>
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[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 211, Eq. (4-72)
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or, separating the term on the left-hand side into two physically distinguishable components &#8212; see equation 44 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; this can be rewritten as,
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<math>\frac{d}{dt} \int\limits_V \rho v_i x_j d^3x - 2 \mathfrak{T}_{ij}</math>
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<math>=</math>
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<math>\delta_{ij}\Pi + \mathfrak{W}_{ij} \, ,</math>
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[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 22, Eq. (47)
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where, by definition,
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&nbsp;
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References
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<math>\mathfrak{T}_{ij}</math>
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<math>\equiv</math>
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<math>\frac{1}{2} \int\limits_V \rho v_i v_j  d^3x </math>
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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is the (ordered) kinetic energy tensor
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 17, Eq. (9)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 212, Eq. (4-74b)
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<math>\Pi</math>
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<math>\equiv</math>
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<math>\int\limits_V P d^3x </math>
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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is &#x2154; of the total thermal (''i.e.,'' random kinetic) energy
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 16, Eq. (7)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 212, Eq. (4-74b)
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<math>\mathfrak{W}_{ij}</math>
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<math>\equiv</math>
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<math>\frac{1}{2} \int\limits_V \rho \Phi_{ij} d^3x </math>
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&nbsp;&nbsp;&nbsp;&nbsp;
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&nbsp;
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 17, Eq. (15)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 68, Eq. (2-126)
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&nbsp;
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<math>=</math>
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<math>- \int\limits_V \rho x_i \frac{\partial \Phi}{\partial x_j} d^3x </math>
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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is the gravitational potential energy tensor
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 18, Eq. (18)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 67, Eq. (2-123)
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Note that, in the definition of the gravitational potential energy tensor, Chandrasekhar has introduced a tensor generalization of the gravitational potential [see his Eq. (14), p. 17], namely,
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<math>- \Phi_{ij} = \mathfrak{B}_{ij}</math>
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<math>=</math>
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<math>G\int\limits_V \rho(\vec{x}^') \frac{ (x_i - x_i^')(x_j - x_j^') }{|\vec{x} - \vec{x}^{~'}|^3} d^3x^' \, ;</math>
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this same potential energy tensor appears explicitly as part of the expression for <math>\mathfrak{W}_{ij}</math> that is presented as Equation (2-126), on p. 67 of [<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>].


The antisymmetric part of this tensor expression gives (see [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] for details),
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<math>\frac{d}{dt} \int\limits_V \rho (v_ix_j - v_j x_i) d^3x</math>
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<math>=</math>
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<math>0 \, ,</math>
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which <font color="#007700">expresses simply the conservation of the angular momentum of the system</font>.  The symmetric part of the tensor expression gives what is generally referred to as (see [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] for details) the,
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<span id="PGE:TVE"><font color="#770000">'''Tensor Virial Equation'''</font></span><br />
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<math>\frac{1}{2} \frac{d^2 I_{ij}}{dt^2}</math>
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<math>=</math>
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<math>2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi \, ,</math>
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[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 23, Eq. (51)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 213, Eq. (4-78)
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<span id="MOItensor">where,</span>
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&nbsp;
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References
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<math>I_{ij}</math>
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<math>\equiv</math>
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<math>\int\limits_V \rho x_i  x_j d^3x </math>
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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is the moment of inertia tensor
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&nbsp;&nbsp;&hellip;&nbsp;&nbsp;
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[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], p. 16, Eq. (4)<br />
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 212, Eq. (4-76)
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