I am reading the mean curvature flow from a thesis. Here, the mean curvature flow is defined as follows:
Let $(M,\varphi_0)$ be an hypersurface. Given $T>0$, the geometric mean curvature flow of $(M,\varphi_0)$, up to $T$, is a family of smooth immersions $\varphi: M\times [0,T)\rightarrow \mathbb{R}^{n+1}$ that satifies
where $H(p,t)$ and $\nu(p,t)$ are respectively the mean curvature and the unit normal of the hypersurface $(M,\varphi_t)$ where $\varphi_t(p)=\varphi(p,t)$.
Later, is said that the above definition is equivalent to a family of smooth vector field $ X:M\times [0,T)\rightarrow \mathbb{R}^{n+1}$ with $X(p,t)\in d\varphi_t(T_pM)$ such that
Actually I can not understand why.
The direction of the flow at $(p,t)$ is $\partial_t\phi(p,t)=:V$, a vector in $\mathbb R^{n+1}$ — or in a suitable tangent space if you have your flow live on a manifold. This vector can be split in two components, one normal to the surface $M_t=\phi(M,t)$ and another one tangent to it. The first formulation says that the normal component of $V$ has size $H$. The second formulation says that the full vector $V$ is $H\nu+X$, where $X$ is tangent to $M_t$. The normal component of $H\nu+X$ has size $H$, so the two descriptions of $V$ are equivalent.
If you are interested in tracking the movement of the surface as a set, then the normal component of $V$ is sufficient information to describe everything. If you want to track the movement of individual points on $M$ in time, then you also need to know the component of motion along the surface.
I hope this illustrates why the two descriptions are equivalent and why you might want to use both.