Pome

Corollary 12.35.11F402CC8C0A7E

Corollary 12.35.11.F402CC8C0A7E For the residual background $\beta _ s=k_ sw_1+a_ sw_2$,

10918993D19D 12.35.12
\begin{equation} \frac{\mathrm d}{\mathrm ds}\sqrt{V_ s} =-\frac{\int _ Y(d w_1+e w_2)\wedge T_ s}{D_\mu (T_ s)}. \end{equation}

Consequently, for every compact segment $[s_0,s_1]$ in that interval,

\[ \sqrt{V_{s_0}}-\sqrt{V_{s_1}} =\int _{s_0}^{s_1} \frac{\int _ Y(d w_1+e w_2)\wedge T_ s}{D_\mu (T_ s)}\, \mathrm ds. \]

Proof. We have $H_ s'=\int em/a_ s^2\, w$, $D_\mu (T_ s)=\sqrt{V_ s}H_ s$ and

\[ \int _ Y(d w_1+e w_2)\wedge T_ s=d+\frac{V_ s}{2}H_ s'. \]

Differentiating $V_ s=2k_ s/H_ s$ gives (12.35.12); integration proves the second identity. This is the explicit first variation of proposition 12.3.2 along the residual family. $\square$

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