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  • ID: 3586990
  • Uploader: redtails »
  • Date: 2020-04-12 15:20
  • Update: 2026-08-01 22:26
  • Size: 931.3 KB .jpg (1600x1200) »
  • Top tagger: redtails (48)
  • Source: twitter.com more
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Media asset #1
What the test will cover
Transform the Laplace operator into polar coordinates.
The Laplace operator is written using the covariant derivative as follows:
This can also be expressed with "div" and "grad" div = divergence of grad = gradient of
Here, we transform grad(f) from Cartesian coordinates into curved coordinates.
As a result, both are transformed in the same way, we can see that grad(f) is represented by a single derivative โˆ‚_i(f).
Therefore, div(A) takes the following form in curved coordinates:
Here, expanding the Christoffel symbol
It can expressed [sic] by the derivative of โˆšg (g is the determinant of g_ij, as the Jacobian squared)
Lastly, substitute the results from B and C into A and use
the explicit forms
We obtain:
Danbooru

Artist's commentary

  • Original
  • Translated
  • โ€• Couverture de test โ€•
    ใฟใ‚“ใชๅฏใฆใ—ใพใ†ใฎใงใ‚ญใƒฌใŸใ‚ณใƒžๅ…ˆ็”Ÿใฏใƒ†ใ‚นใƒˆ็ฏ„ๅ›ฒใ‚’้›ฃใใ—ใฆใ—ใพใ„ใพใ—ใŸ......ใ€‚
    (+30min)
    #่‰ฆใ“ใ‚Œ็‰ˆๆทฑๅคœใฎ็œŸๅ‰ฃใŠ็ตตๆใ60ๅˆ†ไธ€ๆœฌๅ‹่ฒ 
    #่‰ฆใ“ใ‚Œ็‰ˆ็œŸๅ‰ฃใŠ็ตตๆใ60ๅˆ†ไธ€ๆœฌๅ‹่ฒ _20200412
    #ใ‚ณใƒžใƒณใƒ€ใƒณใƒปใƒ†ใ‚นใƒˆ

    โ€• Test cover โ€•
    ใฟใ‚“ใชๅฏใฆใ—ใพใ†ใฎใงใ‚ญใƒฌใŸใ‚ณใƒžๅ…ˆ็”Ÿใฏใƒ†ใ‚นใƒˆ็ฏ„ๅ›ฒใ‚’้›ฃใใ—ใฆใ—ใพใ„ใพใ—ใŸ......ใ€‚
    (+30min)
    #่‰ฆใ“ใ‚Œ็‰ˆๆทฑๅคœใฎ็œŸๅ‰ฃใŠ็ตตๆใ60ๅˆ†ไธ€ๆœฌๅ‹่ฒ 
    #่‰ฆใ“ใ‚Œ็‰ˆ็œŸๅ‰ฃใŠ็ตตๆใ60ๅˆ†ไธ€ๆœฌๅ‹่ฒ _20200412
    #ใ‚ณใƒžใƒณใƒ€ใƒณใƒปใƒ†ใ‚นใƒˆ

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    What the test will cover
    Transform the Laplace operator into polar coordinates.
    The Laplace operator is written using the covariant derivative as follows:
    This can also be expressed with "div" and "grad" div = divergence of grad = gradient of
    Here, we transform grad(f) from Cartesian coordinates into curved coordinates.
    As a result, both are transformed in the same way, we can see that grad(f) is represented by a single derivative โˆ‚_i(f).
    Therefore, div(A) takes the following form in curved coordinates:
    Here, expanding the Christoffel symbol
    It can expressed [sic] by the derivative of โˆšg (g is the determinant of g_ij, as the Jacobian squared)
    Lastly, substitute the results from B and C into A and use
    the explicit forms
    We obtain:

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