First variation formula $ \int_M div _M Y = -\int_M \langle H , Y \rangle $

Question

$M^n\subset \mathbb R^{n+1}$ is a compact without boundary n-dimensional smooth manifold. I see a first variation formula $$ \int_M div _M Y = -\int_M \langle H , Y \rangle $$ $Y$ is a vector field on $M$, $H$ is mean curvature vector. If $Y$ is a tangent vector field , it is obvious right. But if $Y$ have normal component, first, how to define the $div_M Y$. Second, how to show the formula ?

Answer 1

During the proof of the first variation formulae, for a isometric immersion $\psi: M^n \to N^{n+k}$, you have that

$$\frac{d}{dt}dM_t \bigg|_{t=0}= -n\langle H,V\rangle + div(V^T),$$

where $V$ is the variation field of $\psi$. Then the term of the dirvegence is well defined.