Vector cross product with curl

[hotvette]

Homework Statement

Using index-comma notation only, show:
\begin{equation*}
\underline{\bf{v}} \times \text{curl } \underline{\bf{v}}= \frac{1}{2} \text{ grad}(\underline{\bf{v}} \cdot \underline{\bf{v}}) - (\text{grad } \underline{\bf{v}}) \underline{\bf{v}}
\end{equation*}

Homework Equations

\begin{align*}
\text{ curl } \underline{\bf{v}} &= \epsilon_{ijk} v_{j,i} \underline{\bf{e}}_k \\
\underline{\bf{v}} \times \underline{\bf{u}} &= \epsilon_{ijk} v_i u_j \underline{\bf{e}}_k
\end{align*}

The Attempt at a Solution

If I let $\underline{\bf{u}} =\text{ curl } \underline{\bf{v}}$ we get:
\begin{align*}
\underline{\bf{u}} &= \text{ curl } \underline{\bf{v}} =\epsilon_{prq} v_{r,p} \underline{\bf{e}}_q \\
u_j &= \epsilon_{prq} v_{r,p} \underline{\bf{e}}_q \cdot \underline{\bf{e}}_j \\
u_j &= \epsilon_{prq} v_{r,p} \delta_{qj} = \epsilon_{prj} v_{r,p} \\
\underline{\bf{v}} \times \underline{\bf{u}} &= \epsilon_{ijk} v_i u_j \underline{\bf{e}}_k \\
&= \epsilon_{ijk} \epsilon_{prj} v_i v_{r,p} \underline{\bf{e}}_k
\end{align*}
Is this correct so far? Trouble is, I'm not sure what to do next. I'm wondering if trying to simplify $\epsilon_{ijk} \epsilon_{prj}$ would be a fruitful approach. Appreciate hints/tips. Thanks!

 

[Fightfish]

Is this correct so far? Trouble is, I'm not sure what to do next. I'm wondering if trying to simplify $\epsilon_{ijk} \epsilon_{prj}$ would be a fruitful approach.

Looks correct to me. Yup, that approach is the way to go. This is a well-known identity,
$$\epsilon_{ijk} \epsilon_{\ell mk} = \delta_{i\ell} \delta_{jm} - \delta_{im} \delta_{j \ell}$$
for which there are several different ways to prove.

 

[hotvette]

OK, I"m getting closer but (maybe) ran into another snag. Using the following:
\begin{equation*}
\epsilon_{ijk} = \epsilon_{jki} = \epsilon_{kij}
\end{equation*}
we get:
\begin{align*}
\epsilon_{ijk} \epsilon_{prj} &= \epsilon_{kij} \epsilon_{prj} = \delta_{ip} \delta_{jr} - \delta_{ir} \delta_{jp} \\
\underline{\bf{v}} \times \text{curl }\underline{\bf{v}} &=
(\delta_{ip} \delta_{jr} - \delta_{ir} \delta_{jp})(v_i v_{r,p}) \underline{\bf{e}}_k \\
&= v_i v_{r,p} \delta_{ip} \delta_{jr} \underline{\bf{e}}_k - v_i v_{r,p} \delta_{ir} \delta_{jp}\underline{\bf{e}}_k \\
\end{align*}
Looking at the LHS and comparing with the problem statement, we have:
\begin{align*}
v_i v_{r,p} \delta_{ip} \delta_{jr} \underline{\bf{e}}_k &= v_p v_{r,p} \delta_{jr} \underline{\bf{e}}_k \\
\frac{1}{2} \text{grad}(\underline{\bf{v}} \cdot \underline{\bf{v}}) &= \frac{1}{2} (v_i v_i)_{,j} \underline{\bf{e}}_j
= \frac{1}{2} ({v_i}^2)_{,j} \underline{\bf{e}}_j = v_j \underline{\bf{e}}_j
\end{align*}
Which means that $v_p v_{r,p} \delta_{jr} \underline{\bf{e}}_k$ needs to reduce to $v_j \underline{\bf{e}}_j$.
Providing the Kronecker delta can alter partial derivative indices(which I'm not sure), we get:
\begin{align*}
v_p v_{r,p} \delta_{jr} \underline{\bf{e}}_k &= v_p v_{j,p} \underline{\bf{e}}_k \\
&= v_j \underline{\bf{e}}_j \text{ ??}
\end{align*}
It isn't clear to me why the last step is valid. It looks goofy to me because $v_{j,p} = \frac{\partial v_j}{\partial x_p}$ which I think is a matrix of partial derivatives. Appreciate clarification/comment on that one. Now, to tackle the RHS...

 

[vela]

Your claim that $\frac 12(v_i^2)_{,j} = v_j$ isn't correct. If you correct that, you'll see you're essentially done.

 

[Charles Link]

Maybe it's the same thing, but I think the last term should read $v \cdot \nabla v$. (This identity can be found on the cover of J.D. Jackson's Classical Electrodynamics in the form $\nabla (a \cdot b) =a \cdot \nabla b +b \cdot \nabla a + a \times \nabla \times b + b\times \nabla \times a$.)

 

[hotvette]

Hmmm, is:
\begin{equation*}
\tfrac{1}{2} ({v_i}^2)_{,j} = v_i v_{i,j} \text{ ??}
\end{equation*}
If so, then I get:
\begin{align*}
v_p v_{j,p} \underline{\bf{e}}_k &= v_i v_{i,j} \underline{\bf{e}}_j && \text(a) \\
&= v_p v_{p,j} \underline{\bf{e}}_j &&\text{(b) for free index, ok to change}\\
&= v_p v_{j,p} \underline{\bf{e}}_j &&\text{(c) ok to switch indices on partial?}
\end{align*}
But then the basis vectors don't match :(

 

[vela]

We get:
\begin{align*}
\epsilon_{ijk} \epsilon_{prj} &= \epsilon_{kij} \epsilon_{prj} = \delta_{ip} \delta_{jr} - \delta_{ir} \delta_{jp}
\end{align*}

The free indices don't match. You're summing over $j$, so it shouldn't appear on the righthand side.

 

[hotvette]

Ah, thanks for pointing out the error! OK, I now get:
\begin{equation*}
\epsilon_{kij} \epsilon_{prj} = \delta_{kp} \delta_{ir} - \delta_{kr} \delta_{ip}
\end{equation*}
which gives:
\begin{align*}
\underline{\bf{v}} \times \text{curl }\underline{\bf{v}} &= (\delta_{kp} \delta_{ir} - \delta_{kr} \delta_{ip}) v_i v_{r,p} \underline{\bf{e}}_k \\
&= \delta_{kp} \delta_{ir} v_i v_{r,p} \underline{\bf{e}}_k - \delta_{kr} \delta_{ip} v_i v_{r,p} \underline{\bf{e}}_k \\
&= v_r v_{r,k} \, \underline{\bf{e}}_k - v_p v_{k,p} \, \underline{\bf{e}}_k
\end{align*}
For the LHS I now get a match but I still can't get the RHS to match:
\begin{align*}
&\underline{\bf{u}} = \text{grad } \underline{\bf{v}} = v_{i,j} \, \underline{\bf{e}}_i \otimes \underline{\bf{e}}_j && \text{(a)}\\
&\underline{\bf{u}} \underline{\bf{v}} = u_r v_p \, \underline{\bf{e}}_r \otimes \underline{\bf{e}}_p && \text{(b)}\\
&u_r = v_{i,r} \, \underline{\bf{e}}_i && \text{(c)}\\
& (\text{grad } \underline{\bf{v}}) \underline{\bf{v}} = (v_{i,r} \, \underline{\bf{e}}_i) v_p \, \underline{\bf{e}}_r \otimes \underline{\bf{e}}_p && \text{(d)} \\
&= v_p v_{i,r} \, (\underline{\bf{e}}_r \otimes \underline{\bf{e}}_p)\underline{\bf{e}}_i && \text{(e)} \\
&= v_p v_{p,r} \, \underline{\bf{e}}_r = v_p v_{p,k} \, \underline{\bf{e}}_k \ne v_p v_{k,p} \, \underline{\bf{e}}_k && \text{(f)}
\end{align*}
What am I doing wrong?

 

[vela]

Ah, thanks for pointing out the error! OK, I now get:
\begin{equation*}
\epsilon_{kij} \epsilon_{prj} = \delta_{kp} \delta_{ir} - \delta_{kr} \delta_{ip}
\end{equation*}
which gives:
\begin{align*}
\underline{\bf{v}} \times \text{curl }\underline{\bf{v}} &= (\delta_{kp} \delta_{ir} - \delta_{kr} \delta_{ip}) v_i v_{r,p} \underline{\bf{e}}_k \\
&= \delta_{kp} \delta_{ir} v_i v_{r,p} \underline{\bf{e}}_k - \delta_{kr} \delta_{ip} v_i v_{r,p} \underline{\bf{e}}_k \\
&= v_r v_{r,k} \, \underline{\bf{e}}_k - v_p v_{k,p} \, \underline{\bf{e}}_k
\end{align*}

Using what you said in post #6, the first term is $\frac 12 (\vec{v}\cdot \vec{v})_{,k}\hat{e}_k=\frac 12 \nabla(\vec{v}\cdot \vec{v})$. The second term is $(\nabla \cdot \vec{v})\vec{v}$. So you're essentially done. (As Charles pointed out, the last term in the original post is wrong.)

 

[Charles Link]

I don't know this new notation very well= I write out all the partial derivative terms when I do a proof like this, but one suggestion would be to consider my post #5. The term should be $v \cdot \nabla v$, and not $(\nabla v)v$ (or (grad v)v as you wrote it in the OP.)

 

[hotvette]

Ah, I understand now (I didn't know how to interpret post #5). I can now see that it works out if the last term is $\underline{\bf{v}} \cdot \text{ grad} \, \underline{\bf{v}}$. I'll check with my professor. Thanks!