Ytukyg

Let $F_t:Sigma^ntomathbb{R}^{n+1}$ be a smooth family of immersion of a hypersurface that satisfies the inverse curvature flow:
begin{align}
tag1
frac{partial F_t}{partial t}=frac{1}{f}nu.
end{align}
Now, suppose each $Sigma_t:=F_t(Sigma)$ is a star-shaped hypersurface in $mathbb{R}^{n+1}$; that is, there exists a smooth positive function $r$ such that
begin{align}
Sigma_t=left{(r(theta,t),theta):thetainmathbb{S}^nright}
end{align}
Let $(theta^i)$ be a local coordinate system of $mathbb{S}^n$, and denote $sigma_{ij}$ the components of the round metric on $mathbb{S}^n$ with respect to $(theta^i)$. Let $nabla$ be the Levi-Civita connection on $mathbb{S}^n$ with respect to the round metric, and let $|cdot|$ be the norm induced by the round metric. Denote
begin{align}
rho:=sqrt{1+|nablalog r|^2}
end{align}
If we write the flat metric on $mathbb{R}^{n+1}$ by $delta=drotimes dr+r^2g_{mathbb{S}^n}$, then the unit normal vector on $Sigma_t$ can be given by
begin{align}
nu=frac{1}{rho}left(partial_r-(nabla^ilog r)partial_iright)
end{align}

Now, the inverse curvature flow can be reduced to the following equation:
begin{equation}
tag2
frac{partial r}{partial t}=frac{rho}{f}
end{equation}
Indeed, we have the following computation:
begin{align}
frac{1}{f}&=leftlanglefrac{partial F_t}{partial t},nurightrangle \
&=leftlanglefrac{partial r}{partial t}partial_r,frac{1}{rho}left(partial_r-(nabla^ilog r)partial_iright)rightrangle \
&=frac{1}{rho}frac{partial r}{partial t}left[underbrace{langlepartial_r,partial_rrangle}_{=1}-nabla^ilog runderbrace{langlepartial_r,partial_irangle}_{=0}right] \
&=frac{1}{rho}frac{partial r}{partial t}
end{align}

Here comes my question: What is the error in the following
computation: begin{align} frac{partial r}{partial t}&=leftlanglefrac{partial r}{partial t}partial_r,partial_rrightrangle \
&=leftlanglefrac{partial F_t}{partial t},partial_rrightrangle \
&=leftlanglefrac{1}{f}nu,partial_rrightrangle \
&=frac{1}{f}leftlanglefrac{1}{rho}left(partial_r-(nabla^ilog r)partial_iright),partial_rrightrangle \
&=frac{1}{rho f}
end{align}
which obviously does not get the same answer as the computation above?

For reference, I was reading "Flow of Nonconvex Hypersurfaces Into Spheres" by Claus Gerhardt. The equation (2) that I wish to obtain is the equation (1.8) in his paper. In this post, I have reformulated the whole problem in my own language and notations, while also omitted some detail which I think is not relevant to my problem here. I hope my writing is correct and I apologise if this brings any inconvenience to you all.

Thanks in advance for any comment and answer.