Helicity integral in differential forms

[spaghetti3451]

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 $\mathbb{R}^{3}$.

 

[mighty2000]

I am not sure how to incorporate this assumption into my proof. Can someone please help me?

One way to incorporate this assumption is to use the fact that the vorticity 2-form can be written as $\omega^{2} = \omega^{2}_{ij} dx^{i}\wedge dx^{j}$, where $\omega^{2}_{ij}$ are the components of the vorticity tensor. Then, using the fact that $i^{*}\omega^{2} = 0$, we have $i^{*}\omega^{2}_{ij} = 0$ for all $i,j$. This means that the tangential components of the vorticity tensor are zero on the boundary. Using this, you can show that the tangential components of the vorticity vector are also zero on the boundary. Then, you can use this fact in your proof to show that the helicity integral is constant in time.