Multidimensional version of the formula of integration by parts is very frequently used in PDE, hydrodynamics and in many other topics. So it would be nice to have its invariant version. Perhaps the things I am going to write are contained in some big book but I have never seen it anywhere and possibly it these informal notes would be interesting for somebody else.(adsbygoogle = window.adsbygoogle || []).push({});

Let ##D\subset \mathbb{R}^m## stand for open domain such that ##\partial D## is a smooth closed manifold. Assume that in a neighbourhood of ##D## there is a smooth vector field ##v(x)##. Assume also that we are given with differential form ##\omega=\rho(x)dx^1\wedge\ldots\wedge dx^m.##

Theorem. Assume that the form ##\omega## is an invariant form for ##v## i.e. ##L_v\omega=0##, here ##L_v## is the Lie derivative. Then for any smooth functions ##f,g:\overline D\to \mathbb{R}## it follows that

$$\int_Df(L_vg)\omega=\int_{\partial D} fgi_v\omega-\int_Dg(L_vf)\omega.$$ Here ##i_v## is the interior derivative.

Proof. It is easy to see that ##L_v(fg\omega)=(L_vf)g\omega+f(L_vg)\omega.## On the other hand ##L_v(fg\omega)=d(fgi_v\omega)+i_vd(fg\omega).## Gathering these formulas we have

$$d(fgi_v\omega)+i_vd(fg\omega)=(L_vf)g\omega+f(L_vg)\omega.$$ It remains to apply ##\int_D## to both sides of the last equality and note that

$$\int_Dd(fgi_v\omega)=\int_{\partial D}fgi_v\omega,\quad d(fg\omega)=0.$$ QED

1) one can easily restore the detailed conditions of smoothness for all the objects.

2) The arugment remain valid if we replace ##\mathbb{R}^m## with an ##m-##dimensional manifold ##M## and consider proper embedding ##D\subset M##.

3) The conditions of smoothness can actually be weakened up to ##f,g\in H^1(D)##. This also needs for some formal details.

4) If ##D## is a closed manifold we get

$$\int_Df(L_vg)\omega=-\int_Dg(L_vf)\omega.$$ Assume that ##\rho>0## a.e.

So that ##L_v## is ##L^2_\omega(D)-## skew-symmetric operator. It suitable to remember about dynamical systems that preserve measure and the Koopman operator. The Koopman operator ##f\mapsto f(g_v^t(x))## is a unitary operator in ##L^2_\omega(D)-##. Actually this operator is ##e^{tL_v}##. This agrees with skew-symmetricy of ##L_v##.

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# On integration by parts

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