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Lemma (the first variation forula for area, particular case): Let $% X:D\subset %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}$ be a local parametric surface, with $D$ relatively compact domains. Denote by $\xi $ smooth function on $D$ and by $n$ the unit normal vector field of $X$ and set $Y=\xi n$, so the first variation of the area of $X$ is given by $\delta _{\xi n}\left\vert X\right\vert =-2\int_{D}\xi HdA$

Definition1: under the same conditions as above, the volume of $X\left( D\right) ,$denote by $\left\vert CX\right\vert $ is $\frac{1}{3}% \int_{D}\left\langle X,n\right\rangle dA$

Definition2: under the same conditions as above, consider $H_{0}=\frac{% \int_{D}HdA}{\left\vert X\right\vert }$, where $H$ is mean curvature of $X$, and define a functional $J_{D}\left( X\right) $ by $J_{D}\left( X\right) =\left\vert X\right\vert +2H_{0}\left\vert CX\right\vert $.

Now consider a variation of $X_{t}:D\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}$, $t\in \left( -\varepsilon ,\varepsilon \right) $, with $X_{0}=X$

Let $H_{t}$ be the mean curvature of a surface $\left\{ D,X_{t}\right\} $ and $n_{t}$ the unit normal vector field of $X_{t}$, and put $\lambda _{t}=\left\langle \frac{\partial X_{t}}{\partial t},n_{t}\right\rangle $.

Consider $X_{t}$ instead of $X$ in $X_{t}=X+tY$, where $Y$ is vector field in $X\left( D\right) $

Show the details of the second equality

$J_{D}^{\prime }\left( X_{t}\right) =% \frac{d}{dt}J_{D}\left( X_{t}\right) =\int_{D}\left( -2H_{t}+2H_{0}\right) \lambda _{t}dM$

where $dM$ is the area element of $M$.

I know the $J_{D}^{\prime }\left( X_{0}\right) $, but I can not get the expression to $t$ any. I have searched in several books and articles, but people and general does not do this in detail.

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