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Edited September 20, 20241 yr by KJW

[math]\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}[/math]

Edited May 25, 2025May 25 by KJW

[math]|\Phi\!\!>\ ^{\underrightarrow{|\psi_1>}}\ \ |\phi_1\!\!>[/math]

Edited June 29, 2025Jun 29 by KJW

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Edited July 1, 2025Jul 1 by KJW

test

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Edited July 28, 2025Jul 28 by KJW

[math]\text{mathbb: } \mathbb{the\ quick\ brown\ fox\ jumped\ over\ the\ lazy\ dog}[/math]

[math]\text{mathrm: } \mathrm{the\ quick\ brown\ fox\ jumped\ over\ the\ lazy\ dog}[/math]

[math]\text{mathcal: } \mathcal{the\ quick\ brown\ fox\ jumped\ over\ the\ lazy\ dog}[/math]

[math]\text{mathfrak: } \mathfrak{the\ quick\ brown\ fox\ jumped\ over\ the\ lazy\ dog}[/math]

[math]\text{mathscr: } \mathscr{the\ quick\ brown\ fox\ jumped\ over\ the\ lazy\ dog}[/math]

[math]\text{mathbb: } \mathbb{THE\ QUICK\ BROWN\ FOX\ JUMPED\ OVER\ THE\ LAZY\ DOG}[/math]

[math]\text{mathrm: } \mathrm{THE\ QUICK\ BROWN\ FOX\ JUMPED\ OVER\ THE\ LAZY\ DOG}[/math]

[math]\text{mathcal: } \mathcal{THE\ QUICK\ BROWN\ FOX\ JUMPED\ OVER\ THE\ LAZY\ DOG}[/math]

[math]\text{mathfrak: } \mathfrak{THE\ QUICK\ BROWN\ FOX\ JUMPED\ OVER\ THE\ LAZY\ DOG}[/math]

[math]\text{mathscr: } \mathscr{THE\ QUICK\ BROWN\ FOX\ JUMPED\ OVER\ THE\ LAZY\ DOG}[/math]

Edited August 3, 2025Aug 3 by KJW

test before

Spoiler Alert!

test box

test after

Edited September 27, 2025Sep 27 by KJW

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Edited October 18, 2025Oct 18 by KJW

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Edited November 4, 2025Nov 4 by KJW

[math]\mathfrak{e}^{\alpha_1 ... \alpha_n} = \mathfrak{e}_{\alpha_1 ... \alpha_n} = \left\{\begin{array}{cl} 1 & \text{if}\ \ \alpha_1 ... \alpha_n\ \ \text{is an EVEN permutation of}\ \ 1 ... n\\-1 & \text{if}\ \ \alpha_1 ... \alpha_n\ \ \text{is an ODD permutation of}\ \ 1 ... n\\0 & \text{if}\ \ \alpha_1 ... \alpha_n\ \ \text{is NOT a permutation of}\ \ 1 ... n\end{array}\right.[/math]

Edited November 4, 2025Nov 4 by KJW

[math]x \overline{x} x[/math]

[math]x \underline{x} x[/math]

[math]x \buildrel \rm abc \over {xyz} x[/math]

[math]x \buildrel \rm {abc} \over {xyz} x[/math]

Edited November 10, 2025Nov 10 by KJW

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Edited November 30, 2025Nov 30 by KJW

[math]{\partial^2 x \over \partial t^2}[/math]

[math]\displaystyle {\partial^2 x \over \partial t^2}[/math]

[math] \dfrac{\partial^2 x}{\partial t^2}[/math]

[math] \frac{\partial^2 x}{\partial t^2}[/math]

[math]\displaystyle \frac{\partial^2 x}{\partial t^2}[/math]

Edited December 10, 2025Dec 10 by KJW

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Edited December 16, 2025Dec 16 by KJW

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Edited December 21, 2025Dec 21 by KJW

[math]\dfrac{\partial \star}{\partial \overline{x}^{\mu}}[/math]

[math]R{}_{p}^{\mskip{0.05 cm}·}{}_{t}^{\mskip{0.05 cm}·}{}_{p}^{\mskip{0.05 cm}·}{}^{p}_{\mskip{0.05 cm}·}{}^{t}_{\mskip{0.05 cm}·}{}_{p}^{\mskip{0.05 cm}·}{}_{p}^{\mskip{0.05 cm}·}[/math]

[math]R{}_l{}_q{}^r{}_s{}_t{}^g{}^v{}^h{}_x[/math]

Edited March 26Mar 26 by KJW

[math]\displaystyle {\large £}[V{}^{u}] \mathfrak{T}{}^{i_1\ .\ .\ .\ i_p}_{j_1\ .\ .\ .\ j_q} = V{}^{u} \partial{}_{u} \mathfrak{T}{}^{i_1\ .\ .\ .\ i_p}_{j_1\ .\ .\ .\ j_q} - \sum_{\phi = 1}^{p} \partial{}_{u} V{}^{i_\phi}\ \mathfrak{T}{}^{i_1\ .\ .\ .\ i_{\phi - 1}\ u\ i_{\phi + 1}\ .\ .\ .\ i_p}_{j_1\ .\ .\ .\ j_q} + \sum_{\phi = 1}^{q} \partial{}_{j_\phi} V{}^{u}\ \mathfrak{T}{}^{i_1\ .\ .\ .\ i_p}_{j_1\ .\ .\ .\ j_{\phi - 1}\ u\ j_{\phi + 1}\ .\ .\ .\ j_q} +\ w\ \partial{}_{u} V{}^{u}\ \mathfrak{T}{}^{i_1\ .\ .\ .\ i_p}_{j_1\ .\ .\ .\ j_q}[/math]

[math]\displaystyle \overline{{\large £} [V{}^{v}] \mathfrak{T}}{}^{r_1\ .\ .\ .\ r_p}_{s_1\ .\ .\ .\ s_q} = {\large £}[V{}^{u}] \mathfrak{T}{}^{i_1\ .\ .\ .\ i_p}_{j_1\ .\ .\ .\ j_q} \left(\prod_{\lambda = 1}^{p} \dfrac{\partial \overline{x}{}^{r_\lambda}}{\partial x{}^{i_\lambda}}\right) \left(\prod_{\lambda = 1}^{q} \dfrac{\partial x{}^{j_\lambda}}{\partial \overline{x}{}^{s_\lambda}}\right) \left|\dfrac{\partial x{}^{j}}{\partial \overline{x}{}^{s}}\right|^w[/math]

Edited April 14Apr 14 by KJW

13 hours ago, KJW said:

£[Vu]Ti1 . . . ipj1 . . . jq=Vu∂uTi1 . . . ipj1 . . . jq−∑ϕ=1p∂uViϕ Ti1 . . . iϕ−1 u iϕ+1 . . . ipj1 . . . jq+∑ϕ=1q∂jϕVu Ti1 . . . ipj1 . . . jϕ−1 u jϕ+1 . . . jq+ w ∂uVu Ti1 . . . ipj1 . . . jq

£[Vv]T¯¯¯¯¯¯¯¯¯¯¯¯¯¯r1 . . . rps1 . . . sq=£[Vu]Ti1 . . . ipj1 . . . jq(∏λ=1p∂x¯¯¯rλ∂xiλ)(∏λ=1q∂xjλ∂x¯¯¯sλ)∣∣∣∂xj∂x¯¯¯s∣∣∣w

This looks intimidating... What sort of beast is it?

Bear in mind this is "The Sandbox" used for testing how the forum behaves, such as LaTeX code. However, the mathematics itself is about the Lie derivative, a tensorial derivative that is different to the covariant derivative and is independent of the connection.

On 4/15/2026 at 8:13 PM, Anton Rize said:

On 4/15/2026 at 6:22 AM, KJW said:

[math]\displaystyle {\large £}[V{}^{u}] \mathfrak{T}{}^{i_1\ .\ .\ .\ i_p}_{j_1\ .\ .\ .\ j_q} = V{}^{u} \partial{}_{u} \mathfrak{T}{}^{i_1\ .\ .\ .\ i_p}_{j_1\ .\ .\ .\ j_q} - \sum_{\phi = 1}^{p} \partial{}_{u} V{}^{i_\phi}\ \mathfrak{T}{}^{i_1\ .\ .\ .\ i_{\phi - 1}\ u\ i_{\phi + 1}\ .\ .\ .\ i_p}_{j_1\ .\ .\ .\ j_q} + \sum_{\phi = 1}^{q} \partial{}_{j_\phi} V{}^{u}\ \mathfrak{T}{}^{i_1\ .\ .\ .\ i_p}_{j_1\ .\ .\ .\ j_{\phi - 1}\ u\ j_{\phi + 1}\ .\ .\ .\ j_q} +\ w\ \partial{}_{u} V{}^{u}\ \mathfrak{T}{}^{i_1\ .\ .\ .\ i_p}_{j_1\ .\ .\ .\ j_q}[/math]

[math]\displaystyle \overline{{\large £} [V{}^{v}] \mathfrak{T}}{}^{r_1\ .\ .\ .\ r_p}_{s_1\ .\ .\ .\ s_q} = {\large £}[V{}^{u}] \mathfrak{T}{}^{i_1\ .\ .\ .\ i_p}_{j_1\ .\ .\ .\ j_q} \left(\prod_{\lambda = 1}^{p} \dfrac{\partial \overline{x}{}^{r_\lambda}}{\partial x{}^{i_\lambda}}\right) \left(\prod_{\lambda = 1}^{q} \dfrac{\partial x{}^{j_\lambda}}{\partial \overline{x}{}^{s_\lambda}}\right) \left|\dfrac{\partial x{}^{j}}{\partial \overline{x}{}^{s}}\right|^w[/math]

This looks intimidating... What sort of beast is it?

On 4/15/2026 at 8:53 PM, KJW said:

However, the mathematics itself is about the Lie derivative, a tensorial derivative that is different to the covariant derivative and is independent of the connection.

I should also remark that the mathematics is stock standard mathematics used in general relativity. That you call it "intimidating" is revealing about your understanding of general relativity. I personally regard the mathematics as beautiful, although actually doing the mathematics, especially the index manipulations, can be quite tedious to do manually.