Consider a family of metrics and functions $(g(t), u(t))$ on $M:= \mathbb{R}^3 \setminus B_1$ satisfying $$ g(0) = g_0, \quad g'(0) = \tilde g, \quad u(0) = u_0, \quad u'(0) = \tilde u$$ where $g_0$, $\tilde g$ are fixed metrics on $M$ and $u_0$, $\tilde u$ are fixed functions on $M$.
Is there an easy way to compute things like:
$$\left.\frac{d}{dt}\right|_{t=0} \Delta_{g(t)}u(t)$$ $$\left.\frac{d}{dt}\right|_{t=0} A_{g(t)}$$ $$\left.\frac{d}{dt}\right|_{t=0} tr_{g(t)} A$$ $$\left.\frac{d}{dt}\right|_{t=0}\nu_{g(t)} \cdot \nabla_{g(t)}u(t)$$ $$\left.\frac{d}{dt}\right|_{t=0} Hess_{g(t)} u(t) $$ where $A_{g(t)}$ is the second fundamental form on the sphere $S_r$ of radius $r$, and $\nu_{g(t)}$ is the normal unit outward vector field on $S_r$ with respect to $g(t)$.
I am having a very hard time doing this. It's getting very messy and I am not trusting my computations. Is there a book or reference that does things like that?
Any help or reference is appreciated.
(You can suppose that $g_0$ takes the from $dr^2 + h(r)^2 \sigma$ where $\sigma$ is the round metric on $S^2$ and $h$ is some positive function. )
