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## Homework Statement

Let ##V^{3}(t)## be a compact region moving with the fluid.

Assume that at ##t=0## the vorticity ##2##-form ##\omega^{2}## vanishes when restricted to the boundary ##\partial V^{3}(0)##; that is, ##i^{*}\omega^{2}=0##, where ##i## is the inclusion of ##\partial V## in ##\mathbb{R}^{3}##.

(This does ##\textit{not}## say that ##\omega^{2}## itself vanishes, rather only that ##\omega({\vec{u}},{\vec{w}})=0## for ##\vec{u}##,##\vec{w}## tangent to ##\partial V^{3}(0)##.)

Then the ##\textbf{helicity}## integral ##\displaystyle{\int_{V(t)}{\vec{v}}\cdot{\vec{\omega}}\ dx\wedge dy\wedge dz}## can be constant in time.

Prove that the helicity integral is constant in time.

## Homework Equations

## The Attempt at a Solution

$$\frac{d}{dt}\int_{V(t)}{\vec{v}}\cdot{\vec{\omega}}\ dx\wedge dy\wedge dz$$

$$= \frac{d}{dt}\int_{V(t)}{\vec{v}}\cdot{\vec{\omega}}\ dx^{1}\wedge dx^{2}\wedge dx^{3}$$

$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\int_{V(t)} {\vec{v}}\cdot{\vec{\omega}}\ \text{vol}^{3}$$

$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\int_{V(t)} \nu^{1}\wedge \omega^{2}$$

$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\int_{V(t)} \nu^{1}\wedge {\rm d}\nu^{1}$$

$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\int_{V(t)} \nu\wedge {\rm d}\nu$$

$$= \frac{1}{o(x)\sqrt{g}(x)}\frac{1}{2} \frac{d}{dt}\int_{W(t)} (\nu\wedge {\rm d}\nu)+(\nu\wedge {\rm d}\nu)$$

$$= \frac{1}{o(x)\sqrt{g}(x)}\frac{1}{2} \frac{d}{dt}\int_{W(t)} (\nu\wedge {\rm d}\nu)-({\rm d}\nu\wedge \nu)$$

$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\int_{W(t)} {\rm d}(\nu\wedge \nu)$$

$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\oint_{\partial W(t)} \nu\wedge \nu$$

$$= 0.$$

I think this is wrong because I haven't used the assumption that at ##t=0## the vorticity ##2##-form ##\omega^{2}## vanishes when restricted to the boundary ##\partial V^{3}(0)##; that is, ##i^{*}\omega^{2}=0##, where ##i## is the inclusion of ##\partial V## in ##\mathbb{R}^{3}##.

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