# Pulling out a function as a constant in a gradient (domain of linear operators)

Suppose $\Omega_s \subset \mathbb{R}^n$ is a compact subset for each $s \in [0,T]$. I have a linear operator $$p_t^s:H^1(\Omega_t) \to H^1(\Omega_s)$$ which maps functions on $\Omega_t$ to functions on $\Omega_s$, and $p_t^s$ is continuous with continuous inverse $p_s^t$. It is defined via a diffeomorphism $P_t^s:\Omega_s \to \Omega_t$ with $p_t^s f = f \circ P_t^s$.

We can define a modified gradient (the Levi-Civita connection) on $\Omega_t$ by $$\nabla_{t} f := (\nabla f)^T := \nabla f - (\nabla f \cdot N)N$$ where the superscript $T$ denotes projection onto the tangent space of $\Omega_t$ and $N$ is the normal vector on $\Omega_t$. $\nabla$ is the usual gradient on $\mathbb{R}^n$.

Question My question is, is it true that for functions $g \in C^1(\Omega_s)$ $$\nabla_{t} (fg) =g\nabla_{t} (f)?$$ So can I take it out as a constant? I apologise if this question is too easy but I asked on M.SE (http://math.stackexchange.com/questions/265455/question-about-the-definition-of-linear-functions-operators-domains) and did not get a reply.

Essentially, I want to use this condition in an integral over $\Omega_t$ and get a nice cancellation. I tried writing everything out in coordinates (eg. $x$ on $\Omega_t$ and $y$ on $\Omega_y$) and used the diffeomorphism but it's confusing.

Thanks for any help.

-
(1) Its not clear to me that $fg \in \Omega_t$ unless by $fg$ you mean $f p_s^t g$. (2) Have you restricted attention to the case $n = 1$, $\Omega_t = [0,t]$ and worked out the answer in coordinates there? –  Aaron Hoffman Dec 27 '12 at 16:43