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nLab > Latest Changes: Laplace operator

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeMay 6th 2020
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Author: Urs
Format: MarkdownItexstated the definition $\Delta f = \star d \star d f$ and spelled out how this gives the usual component formula: $$ \begin{aligned} \star d \star d f & = \star d \star (\partial_j f) d x^j \\ & = \star d \left( \tfrac{1}{ \color{green} (D-1)! } \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \, g^{ i j} (\partial_j f) \, \epsilon_{ i {\color{green} k_2 \cdots k_{D} } } d x^{ \color{green} k_2 } \wedge \cdots \wedge d x^{ \color{green} k_{D} } \right) \\ & = \star \partial_{ \color{magenta} k_1} \left( \tfrac{1}{ \color{green} (D-1)! } \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \, g^{i j} (\partial_j f) \, \epsilon_{ i {\color{green} k_2 \cdots k_{D} } } d x^{ \color{magenta} k_1 } \wedge d x^{ \color{green} k_2 } \wedge \cdots \wedge d x^{ \color{green} k_{D} } \right) \\ & = \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \underset{ = \det\big( (g_{i j})^{-1} \big) \delta^{ \color{magenta} k_1 }_i }{ \underbrace{ \tfrac{1}{ { \color{orange} D! } { \color{green} (D-1)! } } \epsilon_{ \color{orange} l_1 l_2 \cdots l_D } g^{ { \color{orange} l_1 } { \color{magenta} k_1 } } g^{ { \color{orange} l_2 } { \color{green} k_2 } } \cdots g^{ { \color{orange} l_D} { \color{green} k_D } } \epsilon_{ i {\color{green} k_2 \cdots k_{D} } } } } \, \partial_{ \color{magenta} k_1 } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} (\partial_j f) \right) \\ & = \frac{1}{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \delta^{ \color{magenta} k_1 }_i \partial_{ \color{magenta} k_1 } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} (\partial_j f) \right) \\ & = \frac{1}{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \partial_{i} \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} (\partial_j f) \right) \end{aligned} $$ <a href="https://ncatlab.org/nlab/revision/diff/Laplace+operator/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/Laplace+operator/7">v7</a>, <a href="https://ncatlab.org/nlab/show/Laplace+operator">current</a>

stated the definition $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Δ</mi><mi>f</mi><mo>=</mo><mo>⋆</mo><mi>d</mi><mo>⋆</mo><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">\Delta f = \star d \star d f</annotation></semantics></math>$ and spelled out how this gives the usual component formula:
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo>⋆</mo><mi>d</mi><mo>⋆</mo><mi>d</mi><mi>f</mi></mtd> <mtd><mo>=</mo><mo>⋆</mo><mi>d</mi><mo>⋆</mo><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>j</mi></msub><mi>f</mi><mo stretchy="false">)</mo><mi>d</mi><msup><mi>x</mi> <mi>j</mi></msup></mtd></mtr> <mtr><mtd/> <mtd><mo>=</mo><mo>⋆</mo><mi>d</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mstyle mathcolor="green"><mo stretchy="false">(</mo><mi>D</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo></mstyle></mfrac></mstyle><msqrt><mrow><mo>|</mo><mi>det</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>|</mo></mrow></msqrt><mspace width="0.16667em"/><msup><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msup><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>j</mi></msub><mi>f</mi><mo stretchy="false">)</mo><mspace width="0.16667em"/><msub><mi>ε</mi> <mrow><mi>i</mi><mstyle mathcolor="green"><msub><mi>k</mi> <mn>2</mn></msub><mi>⋯</mi><msub><mi>k</mi> <mi>D</mi></msub></mstyle></mrow></msub><mi>d</mi><msup><mi>x</mi> <mstyle mathcolor="green"><msub><mi>k</mi> <mn>2</mn></msub></mstyle></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mstyle mathcolor="green"><msub><mi>k</mi> <mi>D</mi></msub></mstyle></msup><mo>)</mo></mrow></mtd></mtr> <mtr><mtd/> <mtd><mo>=</mo><mo>⋆</mo><msub><mo>∂</mo> <mstyle mathcolor="magenta"><msub><mi>k</mi> <mn>1</mn></msub></mstyle></msub><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mstyle mathcolor="green"><mo stretchy="false">(</mo><mi>D</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo></mstyle></mfrac></mstyle><msqrt><mrow><mo>|</mo><mi>det</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>|</mo></mrow></msqrt><mspace width="0.16667em"/><msup><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msup><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>j</mi></msub><mi>f</mi><mo stretchy="false">)</mo><mspace width="0.16667em"/><msub><mi>ε</mi> <mrow><mi>i</mi><mstyle mathcolor="green"><msub><mi>k</mi> <mn>2</mn></msub><mi>⋯</mi><msub><mi>k</mi> <mi>D</mi></msub></mstyle></mrow></msub><mi>d</mi><msup><mi>x</mi> <mstyle mathcolor="magenta"><msub><mi>k</mi> <mn>1</mn></msub></mstyle></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mstyle mathcolor="green"><msub><mi>k</mi> <mn>2</mn></msub></mstyle></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mstyle mathcolor="green"><msub><mi>k</mi> <mi>D</mi></msub></mstyle></msup><mo>)</mo></mrow></mtd></mtr> <mtr><mtd/> <mtd><mo>=</mo><msqrt><mrow><mo>|</mo><mi>det</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>|</mo></mrow></msqrt><munder><munder><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mrow><mstyle mathcolor="orange"><mi>D</mi><mo>!</mo></mstyle><mstyle mathcolor="green"><mo stretchy="false">(</mo><mi>D</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo></mstyle></mrow></mfrac></mstyle><msub><mi>ε</mi> <mstyle mathcolor="orange"><msub><mi>l</mi> <mn>1</mn></msub><msub><mi>l</mi> <mn>2</mn></msub><mi>⋯</mi><msub><mi>l</mi> <mi>D</mi></msub></mstyle></msub><msup><mi>g</mi> <mrow><mstyle mathcolor="orange"><msub><mi>l</mi> <mn>1</mn></msub></mstyle><mstyle mathcolor="magenta"><msub><mi>k</mi> <mn>1</mn></msub></mstyle></mrow></msup><msup><mi>g</mi> <mrow><mstyle mathcolor="orange"><msub><mi>l</mi> <mn>2</mn></msub></mstyle><mstyle mathcolor="green"><msub><mi>k</mi> <mn>2</mn></msub></mstyle></mrow></msup><mi>⋯</mi><msup><mi>g</mi> <mrow><mstyle mathcolor="orange"><msub><mi>l</mi> <mi>D</mi></msub></mstyle><mstyle mathcolor="green"><msub><mi>k</mi> <mi>D</mi></msub></mstyle></mrow></msup><msub><mi>ε</mi> <mrow><mi>i</mi><mstyle mathcolor="green"><msub><mi>k</mi> <mn>2</mn></msub><mi>⋯</mi><msub><mi>k</mi> <mi>D</mi></msub></mstyle></mrow></msub></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mi>det</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><msubsup><mi>δ</mi> <mi>i</mi> <mstyle mathcolor="magenta"><msub><mi>k</mi> <mn>1</mn></msub></mstyle></msubsup></mrow></munder><mspace width="0.16667em"/><msub><mo>∂</mo> <mstyle mathcolor="magenta"><msub><mi>k</mi> <mn>1</mn></msub></mstyle></msub><mrow><mo>(</mo><msqrt><mrow><mo>|</mo><mi>det</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>|</mo></mrow></msqrt><msup><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msup><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>j</mi></msub><mi>f</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd/> <mtd><mo>=</mo><mfrac><mn>1</mn><msqrt><mrow><mo>|</mo><mi>det</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>|</mo></mrow></msqrt></mfrac><msubsup><mi>δ</mi> <mi>i</mi> <mstyle mathcolor="magenta"><msub><mi>k</mi> <mn>1</mn></msub></mstyle></msubsup><msub><mo>∂</mo> <mstyle mathcolor="magenta"><msub><mi>k</mi> <mn>1</mn></msub></mstyle></msub><mrow><mo>(</mo><msqrt><mrow><mo>|</mo><mi>det</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>|</mo></mrow></msqrt><msup><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msup><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>j</mi></msub><mi>f</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd/> <mtd><mo>=</mo><mfrac><mn>1</mn><msqrt><mrow><mo>|</mo><mi>det</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>|</mo></mrow></msqrt></mfrac><msub><mo>∂</mo> <mi>i</mi></msub><mrow><mo>(</mo><msqrt><mrow><mo>|</mo><mi>det</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>|</mo></mrow></msqrt><msup><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msup><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>j</mi></msub><mi>f</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \star d \star d f &amp; = \star d \star (\partial_j f) d x^j \\ &amp; = \star d \left( \tfrac{1}{ \color{green} (D-1)! } \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \, g^{ i j} (\partial_j f) \, \epsilon_{ i {\color{green} k_2 \cdots k_{D} } } d x^{ \color{green} k_2 } \wedge \cdots \wedge d x^{ \color{green} k_{D} } \right) \\ &amp; = \star \partial_{ \color{magenta} k_1} \left( \tfrac{1}{ \color{green} (D-1)! } \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \, g^{i j} (\partial_j f) \, \epsilon_{ i {\color{green} k_2 \cdots k_{D} } } d x^{ \color{magenta} k_1 } \wedge d x^{ \color{green} k_2 } \wedge \cdots \wedge d x^{ \color{green} k_{D} } \right) \\ &amp; = \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \underset{ = \det\big( (g_{i j})^{-1} \big) \delta^{ \color{magenta} k_1 }_i }{ \underbrace{ \tfrac{1}{ { \color{orange} D! } { \color{green} (D-1)! } } \epsilon_{ \color{orange} l_1 l_2 \cdots l_D } g^{ { \color{orange} l_1 } { \color{magenta} k_1 } } g^{ { \color{orange} l_2 } { \color{green} k_2 } } \cdots g^{ { \color{orange} l_D} { \color{green} k_D } } \epsilon_{ i {\color{green} k_2 \cdots k_{D} } } } } \, \partial_{ \color{magenta} k_1 } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} (\partial_j f) \right) \\ &amp; = \frac{1}{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \delta^{ \color{magenta} k_1 }_i \partial_{ \color{magenta} k_1 } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} (\partial_j f) \right) \\ &amp; = \frac{1}{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \partial_{i} \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} (\partial_j f) \right) \end{aligned} </annotation></semantics></math>$
diff, v7, current
- CommentRowNumber2.
- CommentAuthorDavid_Corfield
- CommentTimeMay 6th 2020
- PermaLink
Author: David_Corfield
Format: MarkdownItexI always forget we have these colour effects: ${\bgcolor{red} a}$ ${\color{green} b}$ ${\bgcolor{green} \color{red} c}$

I always forget we have these colour effects:

$<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathbackground="red"><mi>a</mi></mstyle></mrow><annotation encoding="application/x-tex">{\bgcolor{red} a}</annotation></semantics></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathcolor="green"><mi>b</mi></mstyle></mrow><annotation encoding="application/x-tex">{\color{green} b}</annotation></semantics></math>$

$<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle mathbackground="green"><mstyle mathcolor="red"><mi>c</mi></mstyle></mstyle></mrow><annotation encoding="application/x-tex">{\bgcolor{green} \color{red} c}</annotation></semantics></math>$
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeMay 6th 2020
- PermaLink
Author: Urs
Format: MarkdownItexAh, thanks for reminding me that there is the background color option, too: `\bgcolor`. Maybe that can be put to use...

Ah, thanks for reminding me that there is the background color option, too: \bgcolor. Maybe that can be put to use…

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nLab > Latest Changes: Laplace operator