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

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.