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I am a rookie so I hope you know the answer! Okay so from C. Möller and Landau & Lifshitz I have gathered that: $$ \gamma_{ij}=g_{ij}-\frac{g_{0i}g_{0j}}{g_{00}} \quad \gamma^{ij}\gamma_{jk}=\delta^i_k\quad \gamma^{ij}=g^{ij}\quad\gamma = \textbf{det}(\gamma_{ab}) = \frac{1}{\textbf{det}(\gamma^{ab})}\\ \epsilon_{ijk} = \epsilon^{ijk}=\left\{\begin{array}{ll} 1 & \text{if } (i,j,k) \text{ is an even permutation of } (1,2,3) \\ -1 &\text{if } (i,j,k) \text{ is an odd permutation of } (1,2,3) \\ 0 & \text{otherwise} \end{array}\right.\\ \varepsilon_{ijk}=\sqrt{\gamma}\epsilon_{ijk}\quad \varepsilon^{ijk}=\frac{1}{\sqrt{\gamma}}\epsilon^{ijk}\\ [\textbf{curl}(\textbf{v})]^{i}=\varepsilon^{ijk} v_{k,j}=\frac{1}{2}\varepsilon^{ijk}(v_{k,j}-v_{j,k})\quad\textbf{div}(\textbf{v})=\frac{1}{\sqrt{\gamma}}[\sqrt{\gamma} v^i]_{,i} $$

Can you similarly define $$ [\mathbf{curl}(\mathbf{\tilde{v}})]_i =\varepsilon_{ijk}v^{k,j} $$ And thereafter $$ [\mathbf{curl}\circ\mathbf{curl}(\mathbf{\tilde{v}})]^a=\varepsilon^{abc} [\varepsilon_{cde}v^{e,d}]_{,b}\\ [\mathbf{curl}\circ\mathbf{curl}(\mathbf{v})]_a=\varepsilon_{abc} [\varepsilon^{cde}v_{e,d}]^{,b} $$ And if so, what is the partial derivative of $\sqrt{\gamma}$ and $\frac{1}{\sqrt{\gamma}}$? Is it valid to reexpress the equations above as $$ [\mathbf{curl}\circ\mathbf{curl}(\mathbf{\tilde{v}})]^a=\varepsilon^{abc} \varepsilon_{cde}v^{e,d}_{\quad,b}\\ [\mathbf{curl}\circ\mathbf{curl}(\mathbf{v})]_a=\varepsilon_{abc} \varepsilon^{cde}v_{e,d}^{\quad,b} $$ or is that wrong? Realized a commaderivative upstairs is not defined. Would it be expressed with codifferential somehow? (http://en.wikipedia.org/wiki/Ricci_calculus#Raised_and_lowered_indices for summation convention)

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1 Answer

I would answer to your question by collecting some facts on the structures you introduce above.

- A typo

The relations

$$\epsilon_{ijk}=\sqrt{\gamma}\epsilon_{ijk}, $$ $$\epsilon^{ijk}=\frac{1}{\sqrt{\gamma}}\epsilon^{ijk}, $$

are not true, unless $\sqrt{\gamma}=1$.

- On curl on $v=v_ie_i$.

The curl operator on vectors $v=v_ie_i$ in $\mathbb R^3$ gives the vector

$$\left(\operatorname{curl}(v)\right)_i=\epsilon_{ijk}\partial_jv_k; $$

such $\operatorname{curl}(v)$ is equal to your $\boldsymbol{\operatorname{curl}}(v)$ as $\epsilon_{ijk}=\epsilon^{ijk}$. Then

$$\boldsymbol{\operatorname{curl}}\left(\boldsymbol{\operatorname{curl}}(v)\right)_i=\operatorname{curl}(\left(\operatorname{curl}(v)\right)_i=\epsilon_{ijk}\partial_j (\epsilon_{krs}\partial_rv_s)= \epsilon_{ijk}\epsilon_{krs}\partial_j \partial_rv_s; $$

using the relations

$$\epsilon_{ijk}\epsilon_{krs}=\epsilon_{kij}\epsilon_{krs}=\delta_{ir}\delta_{js}-\delta_{is}\delta_{jr},$$

one arrives at

$$\operatorname{curl}(\left(\operatorname{curl}(v)\right)_i= (\delta_{ir}\delta_{js}-\delta_{is}\delta_{jr})\partial_j \partial_rv_s=\partial_s\partial_iv_s-\sum_r\partial^2_rv_i, $$

or

$$\boldsymbol{\operatorname{curl}}(v)=\nabla(\operatorname{div}(v))-\nabla^2(v)$$

- On curl on $\tilde{v}=v^ie_i$.

The curl operator on vectors $v=v^ie_i$ in $\mathbb R^3$, where

$$v^i=g^{in}v_n, $$

is the vector

$$\left(\boldsymbol{\operatorname{curl}}(\tilde{v})\right)_i=\epsilon_{ijk}\partial_j(g^{kn}v_n); $$

then

$$\boldsymbol{\operatorname{curl}}\left(\boldsymbol{\operatorname{curl}}(\tilde{v})\right)_i=\epsilon_{ijk}\partial_j (\epsilon_{krs}\partial_r(g^{rq}v_q))= \epsilon_{ijk}\epsilon_{krs}\partial_j (\partial_rg^{rq}v_q+g^{rq}\partial_rv_q). $$

Using once again the relations $$\epsilon_{ijk}\epsilon_{krs}=\delta_{ir}\delta_{js}-\delta_{is}\delta_{jr}$$ you can arrive at the result you are looking for.

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