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ベクトル解析入門pdf化

 ベクトル解析入門pdf化

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kamakiri1225

May 13, 2021
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  1. 𝛻𝑓 𝛻𝑓 𝑥 𝑦 𝑧 等高線 等高線上では全微分(𝒅𝒇=0) 0 = 𝜕𝑓

    𝜕𝑥 𝑑𝑥 + 𝜕𝑓 𝜕𝑦 𝑑𝑥 𝒚を止めて𝒙に ついての微分 𝒙を止めて𝐲に ついての微分
  2. ϕΫτϧղੳೖ໳ — ॳֶऀͷͨΊޯ഑ grad, ൃࢄ div, ճస rot — ʦஶʧӉ஦ʹೖͬͨΧϚΩϦ

    2021 ೥ 5 ݄ 12 ೔ɹ ver 1.0
  3. ˙໔੹ ຊॻ͸৘ใͷఏڙͷΈΛ໨తͱ͍ͯ͠·͢ɻ ຊॻͷ಺༰Λ࣮ߦɾద༻ɾӡ༻ͨ͜͠ͱͰԿ͕ى͖Α͏ͱ΋ɺͦΕ͸࣮ߦɾద༻ɾӡ༻ͨ͠ਓ ࣗ਎ͷ੹೚Ͱ͋Γɺஶऀ΍ؔ܎ऀ͸͍͔ͳΔ੹೚΋ෛ͍·ͤΜɻ ˙঎ඪ ຊॻʹొ৔͢ΔγεςϜ໊΍੡඼໊͸ɺؔ܎֤ࣾͷ঎ඪ·ͨ͸ొ࿥঎ඪͰ͢ɻ ·ͨຊॻͰ͸ɺ™ɺ®ɺ© ͳͲͷϚʔΫ͸লུ͍ͯ͠·͢ɻ

  4. ·͕͖͑ / ͸͡Ίʹ ຊهࣄͰ͸φϒϥԋࢉࢠΛ࢖ͬͨͱͯ΋ॏཁͳʮޯ഑ gradʯ ʮൃࢄ divʯ ʮճస rotʯ ʹ͍ͭͯͷղઆΛߦ͍·͢ɻॳֶऀ͕ҎԼΛݟͯʮԿͩ͜Ε͸ʁʯͱͳΒͳ͍ͨΊͷ

    ষͰ͢ɻ • ޯ഑ɿgrad f(∇ f) • ൃࢄɿdiv v or ∇ · v • ճసɿrot v or ∇ × v ͜ΜͳํΛର৅ʹ͍ͯ͠·͢ɻ • grad, div, rot ʹۤखҙ͕ࣝ͋Δํ • grad ,div ,rot ͷҙຯ΋ؚΊͯཧղ͍ͨ͠ํ https://takun-physics.net/4487/ – i –
  5. None
  6. ໨࣍ ·͕͖͑ / ͸͡Ίʹ i ୈ 1 ষ φϒϥԋࢉࢠͷجૅΛཧղ͢Δ 3

    ͭͷϝϦοτ 1 1.1 ෺ཧݱ৅Λ਺ࣜͰදݱͰ͖Δ . . . . . . . . . . . . . . . . . . . . . 1 1.2 ෺ཧݱ৅ͷཧղͷॿ͚ʹͳΔ . . . . . . . . . . . . . . . . . . . . . 4 1.3 ද͕ࣜ؆୯ʹͳΔ . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 ୈ 2 ষ εΧϥʔ৔ͱϕΫτϧ৔ 7 2.1 εΧϥʔ৔ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 ϕΫτϧ৔ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 φϒϥԋࢉࢠͱ͸ . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 ୈ 3 ষ ޯ഑ 13 3.1 ޯ഑ɿgradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.1 2 ม਺ z = f(x, y) ͷ৔߹ . . . . . . . . . . . . . . . . . . 13 3.1.2 3 ม਺ w = f(x, y, z) ͷ৔߹ . . . . . . . . . . . . . . . . 21 ୈ 4 ষ ൃࢄ 23 4.1 ൃࢄɿdivergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.1.1 ൃࢄ͕ͳ͍৔߹ . . . . . . . . . . . . . . . . . . . . . . . 24 4.1.2 ൃࢄ͕͋Δ৔߹ . . . . . . . . . . . . . . . . . . . . . . . 25 4.1.3 ٵ͍ࠐΈ . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1.4 ͳͥ ∇ ԋࢉࢠͱͷ಺ੵ͕ൃࢄͳͷ͔Λಋग़͢Δ . . . . . . . 28 ୈ 5 ষ ճస 31 5.1 ճసɿrotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.1.1 ͳͥ ∇ ԋࢉࢠͱͷ֎ੵ͕ճసͳͷ͔Λಋग़͢Δ . . . . . . . 34 – iii –
  7. None
  8. ୈ 1 ষ φϒϥԋࢉࢠͷجૅΛཧ ղ͢Δ 3 ͭͷϝϦοτ φϒϥԋࢉࢠͬͯ͝ଘ͡Ͱ͠ΐ͏͔ʁ ͜ͷষͰ͸φϒϥԋࢉࢠʹ͍ͭͯඞͣཧ ղ͓͍ͯͨ͠ํ͕ྑ͍

    3 ͭͷϝϦοτʹ͍ͭͯղઆΛߦ͍·͢φϒϥԋࢉࢠʢ∇ ˡ͍ͭ͜ʣ͕ͪΐͬͱۤखͳํ΍ɺ˸ͷܭࢉͱͦͷҙຯΛؚΊͯ෮श͍ͨ͠ํΛର ৅ʹɺ࣍ষͰղઆ͢Δޯ഑ɺൃࢄɺճసʹ͍ͭͯཧղ͢Δ͜ͱͰφϒϥԋࢉࢠʹର ͢Δۤखҙࣝ͸ແ͘ͳΔ͜ͱͰ͠ΐ͏ɻ ۩ମతʹφϒϥԋࢉࢠΛཧղ͓͍ͯͨ͠ํ͕ྑ͍ 3 ͭͷϝϦοτΛ·ͱΊΔͱҎԼ ͱͳΓ·͢ɻ • ෺ཧݱ৅Λ਺ࣜͰදݱͰ͖Δ • ෺ཧݱ৅ͷཧղͷॿ͚ʹͳΔ • ද͕ࣜ؆୯ʹͳΔ ˞ඍ෼ɺภඍ෼ɺશඍ෼͕ཧղ͍ͯ͠Δ͜ͱΛલఏͱ͍ͯ͠·͢ɻ ʮͰ͸ɺ͸͡Ί·͠ΐ͏ (^^)/ ʯ https://takun-physics.net/12011/ 1.1 ෺ཧݱ৅Λ਺ࣜͰදݱͰ͖Δ φϒϥԋࢉࢠʹ׳Ε͓ͯ͘ͱྑ͍ϝϦοτͷ 1 ͭ໨͕෺ཧݱ৅Λ਺ࣜͰදݱͨ͠ ࡍʹଟ͘ͷ৔໘Ͱొ৔͢Δ͜ͱͰ͢Ͷɻ෺ཧݱ৅Λهड़͢Δํఔࣜʹ͸φϒϥԋࢉࢠ ∇ Λ࢖ͬͨද͕͍ࣜͬͺ͍͋Γ·͢ɻ͜͜Ͱ͸ɺ෼໺͝ͱʹ෼͚ͯφϒϥԋࢉࢠ ∇ ͕࢖ΘΕΔྫΛ͍͔ͭ͘঺հ͠·͢ɻ – 1 –
  9. ୈ 1 ষ φϒϥԋࢉࢠͷجૅΛཧղ͢Δ 3 ͭͷϝϦοτ ʮࣜͷҙຯΛཧղ͢Δඞཁ͸͋Γ·ͤΜɻ͜Μͳʹ͍ͬͺ͍࢖ΘΕͯΔ Αͬͯ͜ͱΛײ͡औ͍ͬͯͩ͘͞ʯ ྗֶ ෺ମʹՃΘΔྗΛ

    F (r)ɺͦΕʹΑΔϙςϯγϟϧΛ U(r) ͱ͠·͢ɻF (r) ͕อଘ ྗʢϙςϯγϟϧྗʣͰ͋Δ৔߹͸ɺҎԼͷΑ͏ͳؔ܎͕ࣜ੒Γཱͪ·͢ɻ ࣜ 1.14: ϙςϯγϟϧྗ F = −∇U (1) ͜ͷ͕ࣜ੒Γཱͭͱ͍͏͜ͱ͸ɺ ࣜ 1.14: ϙςϯγϟϧྗ ∇ × F = 0 (2) ͕੒Γཱͭͱ͍͏͜ͱͱ౳ՁͰ͢ɻ͜Ε͸φϒϥԋࢉࢠΛ࢖ͬͨϕΫτϧެࣜͰ΋ ༗໊ͳ ∇ × F = ∇ × ∇U = 0 ͕੒Γཱ͔ͭΒͰ͢ɻ ͜ͷ࣌఺ͰφϒϥԋࢉࢠΛԡ͓͔͑ͯ͞ͳ͍ͱਏ͘ͳ͖ͬͯ·͢Ͷ (’ Т’) ϊ ి࣓ؾֶ ి࣓ؾֶʹࢸͬͯ͸جૅࣜͰ͋ΔϚΫε΢Σϧํఔࣜ͸ɺ΄ͱΜͲ͕φϒϥԋࢉࢠ Ͱॻ͔Ε͍ͯ·͢ɻ ࣜ 1.14: ϙςϯγϟϧྗ        ∇ · E = ρ ϵ0 ∇ · B = 0 ∇ × E = −∂B ∂t ∇ × B = µ0 j + ϵ0 µ0 ∂E ∂t (3) தֶੜͰशͬͨʮӈͶ͡ͷ๏ଇʯ ʮϑΝϥσʔͷి࣓༠ಋʯͳͲΛࣜͰॻ͖Լ͢ͱ ্هͷΑ͏ͳܗʹ·ͱ·ΔΘ͚Ͱ͢Ͷ (^^) ྔࢠྗֶ ྔࢠྗֶͰ΋φϒϥԋࢉࢠ͸େ׆༂Ͱ͢Ͷɻ ࣜ 1.14: ӡಈྔԋࢉࢠ ˆ p = −iℏ ˆ ∇ – 2 –
  10. 1.1 ෺ཧݱ৅Λ਺ࣜͰදݱͰ͖Δ ͜ͷӡಈྔԋࢉࢠʹΑΓɺ ˆ Hϕ(r) = Eϕ(r) ͔ΒҎԼͷΑ͏ͳγϡϨσΟϯΨʔํ ఔࣜʢ࣌ؒґଘͳ͠ʣ͕ಋ͔Ε·͢ɻ ࣜ

    1.14: γϡϨʔσΟϯΨʔํఔࣜʢ࣌ؒґଘͳ͠ʣ − ℏ2 2m ∇2 + U(r) ϕ(r) = Eϕ(r) (4) γϡϨσΟϯΨʔʹࢸͬͯ͸ ∇2 ͷΑ͏ʹφϒϥԋࢉࢠͷ 2 ৐ؚ͕·Ε͍ͯ· ͢Ͷɻ ྲྀମྗֶ ྲྀମྗֶͷશͯͷجૅࣜʹφϒϥԋࢉࢠؚ͕·Ε͍ͯ·͢ɻ ࣜ 1.14: ࿈ଓͷࣜ ∂ρ ∂t + ∇ · (ρv) = 0 (5) ࣜ 1.14: ӡಈྔอଘ ∂ρv ∂t + ∇ · (ρvv) = ∇ · σ + f (6) ࣜ 1.14: ΤωϧΪʔอଘଇ ∂ρ(e + v·v 2 ) ∂t + ∇ · ρ(e + v · v 2 )v = ∇ · (k∇T) + ∇ · (σ · v) (7) φϒϥԋࢉࢠ͕ݏ͍ͩͱྲྀମྗֶͷࣜΛݟΔͷ΋ݏʹͳΓ·͢ΑͶɻ ఻೤޻ֶ ೤ͷҠಈʹؔͯ͠͸ɺΤωϧΪʔͷҠಈྔ͸Թ౓ࠩʹΑͬͯੜ͍ͯ͡Δͱͯ͠ҎԼ ͷΑ͏ͳϑʔϦΤͷ๏ଇ͕੒Γཱͪ·͢ɻ ෺࣭ͷ೤఻ಋ λ [W/(mɾK)]ɺԹ౓ޯ഑Λ ∇T ͱ͢Δͱɺ೤ྲྀଋ q [W/m2] ͸Ҏ ԼͷࣜͰද͞Ε·͢ɻ ࣜ 1.14: ϑʔϦΤͷ๏ଇ q = −λ∇T (8) – 3 –
  11. ୈ 1 ষ φϒϥԋࢉࢠͷجૅΛཧղ͢Δ 3 ͭͷϝϦοτ ͜Ε͸೤఻ಋʹΑͬͯੜ͡Δ೤ྲྀଋΛදͨࣜ͠Ͱ͢ɻ ͜ͷΑ͏ʹ ∇ Λ࢖ͬͨ෺ཧݱ৅ͷجૅࣜ͸ଟ͘ݟΒΕ·͢ɻ෺ཧݱ৅ͷجૅࣜࣗ

    ମ͕ʮԿ͔෺ཧྔͷมԽྔʢ࣌ؒมԽͩͬͨΓɺۭؒมԽʣ ʯΛද͔ͨࣜͩ͠ΒͰ͢ɻ 1.2 ෺ཧݱ৅ͷཧղͷॿ͚ʹͳΔ φϒϥԋࢉࢠʹ׳Ε͓ͯ͘ͱྑ͍ϝϦοτͷ 2 ͭ໨͕෺ཧݱ৅ͷཧղͷॿ͚ʹͳΔ ͜ͱͰ͢Ͷɻ φϒϥԋࢉࢠͷԋࢉʹ͸ҙຯ͕͋Γ·͢ɻྫ͑͹ɺ ࣜ 1.14: ϙςϯγϟϧྗ F = −∇U (1) Λݟͨͱ͖ʹʮ͋ɺ͜ͷྗ͸อଘྗͩͳʯͱཧղͰ͖·͢ɻ ผͷྫͰ͸ɺ ࣜ 1.14: Ӕͳ͠ྲྀΕ ω = ∇ × v = 0 Ͱ͋Ε͹ʮ͋ɺӔͳ͠ྲྀΕͩͳ (’ Т’) ϊʯͱΘ͔Γ·͢ɻ 1.3 ද͕ࣜ؆୯ʹͳΔ φϒϥԋࢉࢠʹ׳Ε͓ͯ͘ͱྑ͍ϝϦοτͷ 3 ͭ໨͕ද͕ࣜ؆୯ʹͳΔ͜ͱͰ ͢Ͷɻ ྫ͑͹ҎԼͷφϏΤετʔΫεΛ 2 ࣍ݩͷࣜͰॻ͘ͱҎԼͷΑ͏ʹ 2 ͭࣜΛॻ͔ ͳ͚Ε͹ͳΓ·ͤΜɻ ࣜ 1.14: φϏΤετʔΫεํఔࣜ (x ੒෼) ∂u ∂t + u ∂u ∂x + v ∂u ∂y = − 1 ρ ∂p ∂x + ν ∂2u ∂x2 + ∂2u ∂y2 + Fx (9) – 4 –
  12. 1.3 ද͕ࣜ؆୯ʹͳΔ ࣜ 1.14: φϏΤετʔΫεํఔࣜ (y ੒෼) ∂v ∂t +

    u ∂v ∂x + v ∂v ∂y = − 1 ρ ∂p ∂y + ν ∂2v ∂x2 + ∂2v ∂y2 + Fy (10) ͨͩɺ͜ΕΛφϒϥԋࢉࢠΛ࢖͍ɺ଎౓ϕΫτϧΛ v ͱॻ͘͜ͱͰ 1 ͭͷࣜͰॻ͘ ͜ͱ͕Ͱ͖·͢ɻ ࣜ 1.14: φϏΤετʔΫεํఔࣜ ∂v ∂t + (v · ∇)v = − 1 ρ ∇p + ν∇2v + F (11) ˞φϏΤετʔΫεํఔࣜ͸੒෼ͷ਺ͷ͕ࣜ͋Γ·͢ɻ ˞ F ɿ֎ྗʢఆϕΫτϧʣ ࠓճ͸ 2 ࣍ݩΛྫʹऔΓ·͕ͨ͠ɺ3 ࣍ݩͷ৔߹͸͸ 3 ͭํఔࣜΛॻ͘ඞཁ͕͋Γ ·͢ɻ͔͠͠ɺφϒϥԋࢉࢠΛ࢖ͬͯॻ͚͹ 2 ࣍ݩͷ࣌ͱಉ༷ʹεοΩϦͱҰͭͷࣜ (11) Ͱॻ͘͜ͱ͕Ͱ͖·͢ɻ – 5 –
  13. None
  14. ୈ 2 ষ εΧϥʔ৔ͱϕΫτϧ৔ φϒϥԋࢉࢠΛ࢖ͬͨԋࢉΛߦ͏લʹॏཁͳʮεΧϥʔ৔ʯͱʮϕΫτϧ৔ʯʹ ͍ͭͯ؆୯ʹղઆΛ͓͖ͯ͠·͢ɻͳͥɺ ʮεΧϥʔ৔ʯͱʮϕΫτϧ৔ʯʹ͍ͭ ͯղઆ͕ඞཁ͔ͱ͍͏ͱҎޙͷ಺༰ͰφϒϥԋࢉࢠΛ࡞༻ͤͨ݁͞Ռ͸εΧϥʔ ৔ͳͷ͔ϕΫτϧ৔ͳͷ͔ʹΑͬͯ෺ཧతͳҙຯ߹͍͕มΘͬͯ͘Δ͔ΒͰ͢ɻ https://takun-physics.net/4487/

    2.1 εΧϥʔ৔ εΧϥʔ৔͸ɺۭؒͷ֤఺ʹରͯ͠਺஋ʢεΧϥʔྔʣ͕༩͑ΒΕ͍ͯΔঢ়ଶͷ͜ ͱͰ͢ɻ ʮݴ༿Ͱઆ໌ͯ͠΋Α͘Θ͔Βͳ͍ͷͰ۩ମྫΛग़͠·͢ɻʯ ྫ͑͹ɺਤ 2.1 ͷΑ͏ͳ 2 ࣍ݩฏ໘ͰͷԹ౓෼෍͕εΧϥʔ৔ʹ౰ͨΓ·͢ɻ ۭؒ࠲ඪ (x, y) ʹରͯ͠Թ౓͸ T(x, y) ͷΑ͏ͳؔ਺Ͱද͢͜ͱ͕Ͱ͖·͢ɻ͜ͷ Α͏ʹ$(x,y)$ʹରͯͨͩ͠Ұͭͷ஋Λ࣋ͭ΋ͷ͕εΧϥʔ৔Ͱ͢ɻ ͦͷଞʹѹྗ΍ີ౓΋εΧϥʔ৔ʹͳΓ·͢ɻಉ༷ʹ 3 ࣍ݩۭؒΛߟ͑Δͱ (x, y, z) ʹରͯͨͩ͠Ұͭͷ஋Λ࣋ͭ΋ͷ͕εΧϥʔ৔ͰɺԹ౓ΛྫʹऔΔͱ $T(x,y,z)$ͷΑ͏ʹॻ͖·͢ɻ 2.2 ϕΫτϧ৔ ϕΫτϧ৔͸ɺۭؒͷ֤఺ʹରͯ͠ϕΫτϧྔ͕༩͑ΒΕ͍ͯΔঢ়ଶͷ͜ͱͰ͢ɻ ϕΫτϧͱ͸ʮ޲͖ʯͱʮେ͖͞ʯΛ࣋ͬͨྔͰ͢ɻ https://takun-physics.net/11444/ – 7 –
  15. ୈ 2 ষ εΧϥʔ৔ͱϕΫτϧ৔ ˛ ਤ 2.1: Թ౓෼෍ ʮݴ༿Ͱઆ໌ͯ͠΋Α͘Θ͔Βͳ͍ͷͰ۩ମྫΛग़͠·͢ɻʯ ྫ͑͹ɺਤ

    2.3 ͷΑ͏ͳ 2 ࣍ݩฏ໘Ͱͷྲྀ଎ϕΫτϧ෼෍͕ϕΫτϧ৔ʹ౰ͨΓ ·͢ɻ ۭؒ࠲ඪ (x, y) ʹରͯ͠ྲྀ଎͸ v = (vx (x, y), vy (x, y)) ͷΑ͏ʹϕΫτϧͷ x ੒ ෼ɺy ੒෼͸֤఺ͷؔ਺Ͱද͢͜ͱ͕Ͱ͖·͢ɻ ಉ ༷ ʹ 3 ࣍ ݩ ۭ ؒ Λ ߟ ͑ Δ ͱ (x, y, z) ʹ ର ͠ ͯ ϕ Ϋ τ ϧ ৔ ͸ v = (vx (x, y, z), vy (x, y, z), vz (x, y, z)) ͷΑ͏ʹॻ͖·͢ɻ 2.3 φϒϥԋࢉࢠͱ͸ φϒϥԋࢉࢠʢ∇ʣͬͯͦ΋ͦ΋Կ͔ͬͯ͜ͱΛͪΐͼͬͱ࿩͓͔ͯ͠ͳ͍ͱɺ ʮͦ – 8 –
  16. 2.3 φϒϥԋࢉࢠͱ͸ ˛ ਤ 2.2: ϕΫτϧ ͕͜Θ͔ΒΜͷͩʯͬͯࢥΘΕͦ͏ͳͷͰɺ࠷ॳʹ࿩͓͖͍ͯͨ͠ͱࢥ͍·͢ɻ • ∇ɿ ʮφϒϥʯͱݺͼ·͢ɻ

    • ∇ = ( ∂ ∂x , ∂ ∂y , ∂ ∂z ) ˞\(x,y,z\) ͕ม਺ ͱͯ΋؆୯ʹ·ͱΊΔͱҎ্Ͱ͢ɻ ཁ͢Δʹɺ∇ ԋࢉࢠ͸ʮ୭͔Λภඍ෼ͨͯͨ͘͠·Βͳ͍΍ͭʯͱ͘Β͍ʹࢥͬͯ ͓͖·͠ΐ͏ɻ ˙ྫ͑͹ɺf(x, y) = 2x2 + y3 ͱ͍͏ x, y Λม਺ʹ࣋ͬͨؔ਺Λ༻ҙ͠·͢ɻ͜ͷ ؔ਺ʹࠨ͔Βʮ୭͔Λภඍ෼ͨͯͨ͘͠·Βͳ͍ԋࢉࢠ (∇ ԋࢉࢠʣΛ࡞༻ͤ͞Δ͜ ͱΛߟ͑·͢ɻ ͱͯ΋ૉ௚ʹܭࢉ͢Δ͚ͩͰ͢ɻ ࣜ 2.2: φϒϥΛ࡞༻ ∇f(x, y) = ∂ ∂x , ∂ ∂y , ∂ ∂z f(x, y) = ∂f(x, y) ∂x , ∂f(x, y) ∂y , ∂f(x, y) ∂z = 4x, 3y2, 0 – 9 –
  17. ୈ 2 ষ εΧϥʔ৔ͱϕΫτϧ৔ ˛ ਤ 2.3: ྲྀ଎ϕΫτϧ ૉ௚ʹܭࢉͯ͠Έ͚ͨͩͰ͢ɻ εΧϥʔ৔ͷؔ਺

    f(x, y) ʹφϒϥԋࢉࢠΛ࡞༻ͤ͞Δͨ݁Ռ͕ϕΫτϧ৔ʹͳΓ ·ͨ͠Ͷɻ ˙΋͏ͻͱͭྫΛݟͯΈ·͠ΐ͏ɻ ྫ͑͹ɺ͋ΔϕΫτϧ৔ v(x, y) = (vx , vy , vz ) = (2x2, y3, 0) ʹʮ୭͔Λภඍ෼͠ ͨͯͨ͘·Βͳ͍ԋࢉࢠ (∇ ԋࢉࢠʣΛ࡞༻ͤ͞Δ͜ͱߟ͑·͢ɻ ࠓ౓͸಺ੵͷܗͰ࡞༻ͤ͞·͢ɻ ࣜ 2.2: φϒϥͷ಺ੵ ∇ · v(x, y) = ∂vx ∂x + ∂vy ∂y + ∂vz ∂z = 4x + 3y2 ͱͳΓ·͢ɻ ϕΫτϧ৔ͷ v(x, y) ʹφϒϥԋࢉࢠͷ಺ੵΛ࡞༻ͤ͞Δͨ݁Ռ͕εΧϥʔ৔ʹͳ Γ·ͨ͠Ͷɻ – 10 –
  18. 2.3 φϒϥԋࢉࢠͱ͸ ·ͣ͸ૉ௚ʹʮ∇ ԋࢉࢠΛ࢖͏ͱ·ΔͰϕΫτϧͰֶशͨ͠಺༰ͬΆ͘ܭࢉͰ͖Δ ͳʯͬͯײͯ͡׳Ε͍͚ͯ͹Α͍ͷ͔ͱࢥ͍·͢ɻ ͔͠͠ɺ෺ཧ਺ֶͷϕΫτϧղੳͰφϒϥԋࢉࢠΛษڧ͢Δͱʮgrad, div,rotʯ΋ Ұॹʹश͏͜ͱʹͳΓ·͢ɻ͜͜Ͱ͸΋͏গ͠φϒϥԋࢉࢠΛ࢖ͬͨ୯ͳΔԋࢉࢠͷ ܭࢉʹͱͲ·Βͳ͍ҙຯΛ࣋ͬͨૢ࡞Ͱ͋Δ͜ͱΛઆ໌͍ͨ͠ͱࢥ͍·͢ɻ –

    11 –
  19. None
  20. ୈ 3 ষ ޯ഑ ͜ΜͳํΛର৅ʹ͍ͯ͠·͢ɻhttps://takun-physics.net/4487/ 3.1 ޯ഑ɿgradient ·ͣ͸φϒϥԋࢉࢠΛ࢖ͬͨ؆୯ͳྫ͸ʮޯ഑ʯͰ͢ɻ ͳͥ grad

    ͱॻ͘ͷ͔ͱ͍͏ͱޯ഑ͷӳޠ͕ gradient ͔ͩΒͦͷ๯಄ 3 จࣈΛ࢖ͬ ͯ grad ͱॻ͘ͷͰ͢ɻ ʮޯ഑ʯʹؔ͢Δॻ͖ํ͸ɺgrad f ͩͬͨΓ ∇f ͩͬͨΓ͠·͕͢ɺ͋Δؔ਺ f ʹ ∇ ԋࢉࢠΛ࡞༻ͤ͞Δૢ࡞ʹΑͬͯͰ͖·͢ɻ ͜Μͳײ͡Ͱॻ͖·͢ɻ ࣜ 3.7: gradient grad f = ∇f = ∂f ∂x , ∂f ∂y ؆୯ͳྫͰݟͯΈ·͠ΐ͏ɻ ͜ΕΛݟΕ͹ɺ∇ ԋࢉࢠΛ͋Δؔ਺ f ʹ࡞༻ͤ͞Δૢ࡞͕ʮޯ഑ʯͰ͋Δ͜ͱ͕Θ ͔Δ͔ͱࢥ͍·͢ɻ 3.1.1 2 ม਺ z = f(x, y) ͷ৔߹ 2 ม਺ (x, y) ͷ৔߹ͷશඍ෼Λߟ͑·͠ΐ͏ɻ – 13 –
  21. ୈ 3 ষ ޯ഑ ˛ ਤ 3.1: 2 ม਺ʹ͓͚Δޯ഑ ͜͜Ͱ

    (x, y) ฏ໘্Ͱ (x, y) ͔Β (x + ∆x, y + ∆y) ΁มԽͨ͠ͱ͖ʹߴ͞ํ޲ͷ ∆f = f(x + ∆x, x + ∆y) − f(x, y) ͸ͲͷΑ͏ʹॻ͚Δ͔Λߟ͑·͢ɻ ֆͰඳ͘ਤ 3.2 ͷͱ͜Μͳײ͡ɻ – 14 –
  22. 3.1 ޯ഑ɿgradient ˛ ਤ 3.2: dx ͱ dy ʹ͓͚Δ f(x,

    y) ͷมԽྔ ͜ΕΛͻͱͭͻͱͭߟ͑ͯΈ·͢ɻ – 15 –
  23. ୈ 3 ষ ޯ഑ y ํ޲ΛࢭΊͯ x ํ޲͚ͩͷมԽΛߟ͑·͢ɻ ˛ ਤ

    3.3: y ํ޲ΛࢭΊͯ x ํ޲มԽʹ͓͚Δ f(x, y) ͷมԽྔ ࣜ 3.7: y ʹ͓͚Δ f ͷมԽྔ f(x + ∆x, y) − f(x, y) ∆x ∆x → ∆x→0 = ∂f ∂x dx ͱͳΓ·͢ɻ ˞ ∂f ∂x ͸ y Λݻఆͯ͠ x ͰͷΈඍ෼Λߦ͏͜ͱΛҙຯ͠·͢ʢx ʹΑΔภඍ෼ʣ – 16 –
  24. 3.1 ޯ഑ɿgradient x ํ޲ΛࢭΊͯ y ํ޲͚ͩͷมԽΛߟ͑·͢ɻ ˛ ਤ 3.4: x

    ํ޲ΛࢭΊͯ y ํ޲มԽʹ͓͚Δ f(x, y) ͷมԽྔ ࣜ 3.7: x ʹ͓͚Δ f ͷมԽྔ f(x, y + ∆y) − f(x, y) ∆y ∆y → ∆y→0 = ∂f ∂y dy ͱͳΓ·͢ɻ ˞ ∂f ∂y ͸ x Λݻఆͯ͠ y ͰͷΈඍ෼Λߦ͏͜ͱΛҙຯ͠·͢ʢx ʹΑΔภඍ෼ʣ Ҏ্ΑΓ ∆f = f(x + ∆x, x + ∆y) − f(x, y) ͸ᶃ + ᶄͰ͋Δ͜ͱ͕Θ͔Γ·͢ɻ – 17 –
  25. ୈ 3 ষ ޯ഑ ˛ ਤ 3.5: f(x, y) ͷมԽྔ

    ඍ෼͸্ͷදࣜʹͳΓ·͢ɻ ࣜ 3.7: શඍ෼ df = ∂f ∂x dx + ∂f ∂y dy ͜ΕΛ͜ͷΑ͏ʹ֤੒෼ʹ෼ղͯ͠಺ੵͷܗͰॻ͘ͱɺ ࣜ 3.7: x ʹ͓͚Δ f ͷมԽྔ df = ∂f ∂x , ∂f ∂y · (dx, dy) ͱͳΓ·͢ɻ ͓ͬͱɺ ʮ ∂f ∂x , ∂f ∂y ͕ग़͖ͯͨͰ͸͋Γ·ͤΜ͔ʯͱ͍͏͜ͱʹͳΓ·͢ɻ ͜ΕΛɺ – 18 –
  26. 3.1 ޯ഑ɿgradient ࣜ 3.7: x ʹ͓͚Δ f ͷมԽྔ grad f

    = ∇f = ∂f ∂x , ∂f ∂y ͱॻ͍ͯޯ഑Λද͢ͷͰ͢ɻ Ͱ͸ɺͳͥޯ഑ͱݺͿͷ͔ɾɾɾɾwikipedia ʹ͸ʮޯ഑;gradientʯʹ͍ͭͯɺ ˝ ޯ഑ (grad) ͱ͸ ϕΫτϧղੳʹ͓͚ΔεΧϥʔ৔ͷޯ഑(gradient; άϥσΟΤϯτ ʣ ͸ ɺ ֤఺› ›ʹ͓͍ͯͦͷεΧϥʔ৔ͷมԽ཰͕࠷େͱͳΔํ޲΁ͷมԽ཰ͷ஋Λେ͖͞ʹ΋› ›ͭϕΫτϧΛରԠͤ͞ΔϕΫτϧ৔Ͱ͋Δ ɻ ͜ͷΑ͏ʹॻ͍͍ͯ·͢ɻ εΧϥʔ৔ͷมԽ཰͕࠷େͱͳΔํ޲΁ͷมԽ཰ͱॻ͍ͯ͋Γ·͢ɻ ʮݴ༿Ͱઆ໌ͯ͠Πϝʔδͭ͘ਓ΋͍ͳ͍Ͱ͠ΐ͏ʂ ʯ ͱ͍͏Θ͚Ͱ ∇f Λޯ഑ͱ͍͏ҙຯ͕Θ͔ΔΑ͏ʹ gradf ͷҙຯΛߟ͑ͯΈ·͢ɻ ౳ߴઢʹରͯ͠ਨ௚ͳํ޲͕ ∇f ͷํ޲ ઌ΄Ͳॻ͍͕ͨࣜ಺ੵͷܗͰॻ͘͜ͱ͕Ͱ͖ͨͷͰɺҎԼͷΑ͏ʹॻ͖·͠ΐ͏ɻ ࣜ 3.7: x ʹ͓͚Δ f ͷมԽྔ df = |grad f| |∆r| cos θ • x, y ฏ໘্ͷඍখมԽΛ r = (dx, dy) ͱॻ͖·ͨ͠ɻ • grad f ͱ ∆r ͷͳ֯͢౓Λ cos θ ͱ͍ͯ͠·͢ɻ ʮ౳ߴઢʹରͯ͠ਨ௚ͳํ޲͕ ∇ f ͷํ޲ʯͱ͸Ͳ͏͍͏ҙຯ͔ߟ͑ͯΈ·͠ΐ͏ɻ ౳ߴઢΛඳ͍ͯߟ͑ͯΈ·͢ɻ∆r ͷಈ͔͠ํΛ࣍ͷ 2 ௨Γͷ৔߹Ͱߟ͑ͯΈ ·͢ɻ • ∆r Λ౳ߴઢͷํ޲ʹಈ͔͢৔߹ • ∆r Λ df ͕࠷େʹͳΔΑ͏ʹಈ͔͢৔߹ ˙ ∆r Λ౳ߴઢͷํ޲ʹಈ͔͢৔߹ʹ͍ͭͯ – 19 –
  27. ୈ 3 ষ ޯ഑ ˛ ਤ 3.6: grad ͸౳ߴઢʹରͯ͠ਨ௚ ∆r

    Λ౳ߴઢͷํ޲ʹಈ͔͢ͱɺ$df$ͷߴ͕͞มΘΒͳ͍ͨΊ$df=0$ͱͳΓ·͢ɻ ͳͷͰɺ಺ੵͷఆ͔ٛΒ θ = 90 ͱͳΓɺgrad f ͱ ∆r ͸௚ަ͍ͯ͠Δ͜ͱ͕Θ͔Γ ·͢ɻ ͭ·Γɺgrad f ͸౳ߴઢରͯ͠ਨ௚Ͱ͋Δ͜ͱ͕Θ͔Γ·͢ɻ ˙ ∆r Λ df ͕࠷େʹͳΔΑ͏ʹಈ͔͢৔߹ – 20 –
  28. 3.1 ޯ഑ɿgradient ˛ ਤ 3.7: grad ͸ df ͕࠷େมԽ͢Δํ޲ʹ޲͍͍ͯΔ ∆r

    Λ$df$͕࠷େʹͳΔํ޲ʹಈ͔ͨ͠৔߹ɺ ಺ੵͷఆ͔ٛΒ θ = 0 ͱͳΓɺ grad f ͱ ∆r ͕ಉ͡ํ޲Λ޲͍͍ͯΔ͜ͱ͕Θ͔Γ·͢ɻ ͭ·Γɺgrad f ͸ df ͕࠷େʹͳΔํ޲Λ޲͍͍ͯΔ͕Θ͔Γ·͢ɻ Ώ͑ʹɺgrad f ͸εΧϥʔ৔ͷมԽ཰͕࠷େͱͳΔํ޲ͱઆ໌Ͱ͖ΔΘ͚Ͱ͢ɻ Ͱ͸ɺ3 ม਺ͩͬͨΒʁ 3.1.2 3 ม਺ w = f(x, y, z) ͷ৔߹ 3 ม਺ʹͳΔදݱ͢ΔͨΊͷ͕࣠଍Γͳ͍ͷͰ w ͷେ͖͞͸৭ͱ͔Ͱ۠ผ͢Δ͠ ͔ͳ͍Ͱ͢ɻ ઌ΄Ͳʮ2 ม਺Ͱ౳ߴઢ (z = f(x, y) ͷ஋͕ৗʹಉ͡)ʯΛߟ͑ͨΑ͏ʹɺ ʮ3 ม਺ Ͱ౳Ґ໘ (w = f(x, y, z) ͷ஋͕ৗʹಉ͡)ʯ৔߹Ͱͷ ∇ f ͷํ޲Λߟ͑Δ͜ͱ͕Ͱ͖ ·͢ɻ3 ม਺Ͱͷ౳Ґ໘Ͱͷ ∇f ͷํ޲͸ɺ౳Ґ໘ʹରͯ͠ਨ௚ͳํ޲Ͱ͢ɻ – 21 –
  29. ୈ 3 ষ ޯ഑ ˛ ਤ 3.8: 3 ม਺ʹ͓͚Δ grad

    ͸౳Ґ໘ʹରͯ͠ਨ௚ͳํ޲Λ޲͍͍ͯΔ ͜͜·Ͱ཈͓͚͑ͯ͹جຊతͳ͜ͱ͸ OK Ͱ͢ʂ – 22 –
  30. ୈ 4 ষ ൃࢄ ຊهࣄͰ͸φϒϥԋࢉࢠΛ࢖ͬͨͱͯ΋ॏཁͳʮޯ഑ gradʯ ʮൃࢄ divʯ ʮճస rotʯʹ͍ͭͯͷղઆΛߦ͍·͢ɻॳֶऀ͕ҎԼΛݟͯʮԿͩ͜Ε͸ʁʯͱͳΒͳ͍

    ͨΊͷষͰ͢ɻ* ޯ഑ɿgrad f(∇ f) * ൃࢄɿdiv v or ∇ · v * ճసɿrot v or ∇ × v ͜ΜͳํΛର৅ʹ͍ͯ͠·͢ɻ • div ʹۤखҙ͕ࣝ͋Δํ • div ͷҙຯ΋ؚΊͯཧղ͍ͨ͠ํ https://takun-physics.net/4487/ 4.1 ൃࢄɿdivergence ਺ֶه߸Ͱͷൃࢄͷදݱ͸ div ͔ ∇· ͱͳΓ·͢ɻ∇ ԋࢉࢠͱ͋ΔϕΫτϧ৔ v ͱ ͷ಺ੵͰ͋Δͱ͓͖֮͑ͯ·͠ΐ͏ɻ@href{https://takun-physics.net/11522/} ͳͥ div ͱॻ͘ͷ͔ͱ͍͏ͱൃࢄͷӳޠ͕ divergence ͔ͩΒͦͷ๯಄ 3 จࣈΛ ࢖ͬͯ div ͱॻ͘ͷͰ͢ɻ ·ͨɺ∇ = ( ∂ ∂x , ∂ ∂y , ∂ ∂z ) Λ࢖ͬͯɺ͋ΔϕΫτϧͱͷ಺ੵΛͱΔͱͦΕ͸ൃࢄΛҙ ຯ͢Δ͜ͱʹͳΓ·͢ɻ ࣜ 4.16: div div v = ∇ · v ͱॻ͖·͢ɻ φϒϥԋࢉࢠͷ෦෼ʹ͍ͭͯɺ ʮه߸Λ࢖͍ͬͯͯ͸Θ͔Βͳ͍ʯͬͯํ͸ɺ࣮ࡍ ʹ಺ੵΛܭࢉͯ͠΍Δͱɺ ࣜ 4.16: div div v = ∂vx ∂x + ∂vy ∂y + ∂vz ∂z – 23 –
  31. ୈ 4 ষ ൃࢄ ͱͳΔ͜ͱΛ͓͚֮͑ͯ͹ྑ͍Ͱ͠ΐ͏ɻ ͔͠͠ɺ਺ֶه߸ͰൃࢄͱݴΘΕͯ΋ϐϯͱ͜ͳ͍ͷ͕;ͭ͏Ͱ͋Δͱࢥ͍·͢ɻ ·ͣ͸؆୯ʹΠϝʔδΛ಄ʹΠϯϓοτ͢ΔͨΊʹɺ͋ΔϕΫτϧ৔Λྲྀ଎ϕΫτ ϧ v ͱͯ͠઒ͷྲྀΕΛߟ͑ͯɺ

    ʮൃࢄʯʹ͍ͭͯཧղ͍͖͍ͯͨ͠ͱࢥ͍·͢ɻ 4.1.1 ൃࢄ͕ͳ͍৔߹ ൃࢄ͕ͳ͍৔߹ͱ͍͏ɺ ʮൃࢄʯͱ͍͏ݴ༿Λ࢖͏ͱΑ͘Θ͔Βͳ͍͜ͱͰ͠ΐ ͏Ͷɻ ൃࢄ͕ͳ͍৔߹ͱ͍͏ͷ͸ɺ ʮൃࢄʹग़͍ͯͬͨྔʢྲྀग़ʣʔೖ͖ͬͯͨྔʢྲྀೖʣ ʯ ͸ 0 ͱ͍͏৔߹Λҙຯ͍ͯ͠·͢ɻ େࣄͳͷ͸ɺൃࢄ͸ਖ਼ຯʹग़͍ͯͬͨྔͱ͍͏͜ͱͰ͢ɻ ྲྀྔ͸ग़͍ͯ͘ΜͰ͚͢Ͳɺೖͬͯ͘Δྔ΋߹Θͤͯɺ ʮ࣮ࡍͲΕ͚ͩग़͍ͯͬͨ ͷ͔ʁʯ͕ൃࢄͷҙຯͰ͢ɻ ˛ ਤ 4.1: ྲྀೖͱྲྀग़ɿൃࢄແ͠ ྫ͑͹্ͷֆͷΑ͏ʹ 1 ࣍ݩͷྲྀ଎͕ҰఆͷྲྀΕ͕͋Δ৔߹Λߟ͑·͢ɻ ͜ͷ৔߹͸ɺ྘ͷ൒ಁ໌ʹೖ͖ͬͯͨਫͷྔͱग़͍ͯͬͨਫͷྔ͸ಉ͡Ͱ͢ΑͶɻ ͔ͩΒɺ ʮൃࢄʹग़͍ͯͬͨྔʢྲྀग़ʣʔೖ͖ͬͯͨྔʢྲྀೖʣ ʯ͸ 0 ͱ͍͏͜ͱʹ ͳΓ·͢ɻ – 24 –
  32. 4.1 ൃࢄɿdivergence ൃࢄه߸Ͱॻ͘ͱɺ ࣜ 4.16: ൃࢄ͕ͳ͍৔߹ div v = 0

    Ͱ͢ɻ φϒϥԋࢉࢠΛ༻͍Δͱɺ ࣜ 4.16: ൃࢄ͕ͳ͍৔߹ (φϒϥԋࢉࢠΛ࢖ͬͨදݱ) ∇ · v = 0 Ͱ͢ɻ ࣮ࡍɺྲྀΕ͸ҰఆͳͷͰ͔͢Β dv dx = 0 Ͱ͋Γ·͢ɻ 3 ࣍ݩͰͷൃࢄແ͠ͷ৔߹΋ɺ1 ࣍ݩͷ৔߹ͱಉ༷ʹɺ ࣜ 4.16: 3 ࣍ݩɿൃࢄ͕ͳ͍৔߹ (φϒϥԋࢉࢠΛ࢖ͬͨදݱ) ∇ · v = ∂vx ∂x + ∂vy ∂y + ∂vz ∂z = 0 ͱॻ͘͜ͱ͕Ͱ͖·͢ɻ 4.1.2 ൃࢄ͕͋Δ৔߹ ࠓ౓͸ൃࢄ͕͋Γͷ৔߹ʹ͍ͭͯߟ͑ͯΈ·͠ΐ͏ɻൃࢄ͕͋Δ৔߹ͱ͍͏ͷ͸ɺ ʮൃࢄʹग़͍ͯͬͨྔʢྲྀग़ʣʔೖ͖ͬͯͨྔʢྲྀೖʣ ʯ͸ 0 Ͱ͸ͳ͍ͱ͍͏͜ͱΛҙ ຯ͍ͯ͠·͢ɻྲྀೖ͖ͯͨ͠ྔʹରͯ͠ʮ༙͖ग़͠ʯ΋͘͠͸ʮٵ͍ࠐΈʯ͕͋Δͱ ͍͏͜ͱͰ͢ɻ – 25 –
  33. ୈ 4 ষ ൃࢄ ༙͖ग़͠ ˛ ਤ 4.2: ྲྀೖͱྲྀग़ɿൃࢄ͋Γ ਤ

    4.2 ͷֆͷΑ͏ʹӈʹ͍͘΄Ͳʢ\(x\) ͷ૿Ճͱͱ΋ʹ) ྲྀ଎͕૿͍͍͑ͯͬͯ Δ৔߹ͷྲྀΕͰ͸ɺ྘ͷಁ໌ʹೖ͖ͬͯͨਫͷྔΑΓग़͍ͯ͘ਫͷྔͷํ͕ଟ͍ͷ Ͱɺਖ਼ຯͷग़͍ͯͬͨྔͱ͍͏ͷ͸ 0 ΑΓେ͖͍Ͱ͢ɻ ͜ΕΛ༙͖ग़͠ͱݴ͍·͢ɻ ࣜ 4.16: ൃࢄ͋Γ (༙͖ग़͠) div v > 0 Ͱ͢ɻ φϒϥԋࢉࢠΛ༻͍Δͱɺ ࣜ 4.16: ൃࢄ͋Γ (༙͖ग़͠) ∇ · v > 0 ͱॻ͘͜ͱ͕Ͱ͖·͢ɻ – 26 –
  34. 4.1 ൃࢄɿdivergence 4.1.3 ٵ͍ࠐΈ ༙͖ग़͠ͷٯͷٵ͍ࠐΈͷ৔߹Λߟ͑·͠ΐ͏ɻ ˛ ਤ 4.3: ྲྀೖͱྲྀग़ɿൃࢄ͋Γ ਤ

    4.3 ͷΑ͏ʹɺ྘ͷಁ໌ʹೖ͖ͬͯͨਫͷྔྲྀೖ͖ͯͨ͠ྔʹରͯ͠ग़͍ͯ͘ਫ ͷྔͷํ͕ଟগͳ͍৔߹͸ਖ਼ຯͷग़͍ͯͬͨྔͱ͍͏ͷ͸ 0 ΑΓখ͘͞ͳΓ·͢ɻ ͜ΕΛٵ͍ࠐΈͱݴ͍·͢ɻ ࣜ 4.16: ൃࢄ͋Γ (ٵ͍ࠐΈ) div v < 0 φϒϥԋࢉࢠΛ༻͍Δͱɺ ࣜ 4.16: ൃࢄ͋Γ (ٵ͍ࠐΈ) ∇ · v < 0 ͱॻ͘͜ͱ͕Ͱ͖·͢ɻ – 27 –
  35. ୈ 4 ষ ൃࢄ 4.1.4 ͳͥ ∇ ԋࢉࢠͱͷ಺ੵ͕ൃࢄͳͷ͔Λಋग़͢Δ Ͱ͸ͳͥ ∇

    ԋࢉࢠͷ಺ੵ͕ൃࢄΛҙຯ͍ͯ͠Δͷ͔Λߟ͍͑ͨͱࢥ͍·͢ɻ x, y, z ํ޲ʹਨ௚ͳ໘͔Βग़͍ͯ͘ਖ਼ຯͷྲྀྔΛߟ͑Δ͜ͱͰൃࢄΛཧղ͍ͨ͠ͱ ࢥ͍·͢ɻ ˛ ਤ 4.4: x ํ޲ʹਨ௚ͳ໘͔Βग़͍ͯ͘ਖ਼ຯͷྔ ͯ͞ɺӈͷ໘͔Βग़͍ͯ͘ྲྀྔͱࠨͷ໘͔Βೖͬͯ͘Δྲྀྔͱ͍͏ͷΛߟ͑·͢ɻ • ग़͍ͯ͘ྲྀྔɿvx (x + dx)dydz • ೖͬͯ͘Δྲྀྔɿvx (x)dydz ˞ dx, dy, dz ͸ඍখྔͱ͍ͯ͠·͢ɻ ͢Δͱਖ਼ຯͷग़͍ͯ͘ྲྀྔ͸ɺ ࣜ 4.16: x ํ޲ʹਨ௚ͳ໘͔Βग़͍ͯ͘ਖ਼ຯͷྔ vx (x + dx)dydz − vx (x)dydz ͱ͍͏͜ͱʹͳΓ·͢ɻ͜ΕΛ΋͏গࣜ͠มܗͯ͠Έ·͠ΐ͏ɻ – 28 –
  36. 4.1 ൃࢄɿdivergence ࣜ 4.16: x ํ޲ʹਨ௚ͳ໘͔Βग़͍ͯ͘ਖ਼ຯͷྔ vx (x+dx)dydz−vx (x)dydz =

    (vx (x+dx)−vx (x))dydz = vx (x + dx) − vx (x) dx dxdydz ͜͜Ͱ dx ˠ 0 ͱ͢Δͱɺvx(x+dx)−vx(x) dx = dvx dx ͔ͩΒɺ x ํ޲ʹਨ௚ͳ໘͔Βग़͍ͯ͘ਖ਼ຯͷྲྀྔ ࣜ 4.16: x ํ޲ʹਨ௚ͳ໘͔Βग़͍ͯ͘ਖ਼ຯͷྔ dvx dx dxdydz ͱͳΓ·͢ɻ ಉ༷ͷखॱΛʮy ํ޲ʹਨ௚ͳ໘ʯ ʮz ํ޲ʹਨ௚ͳ໘ʯͷਖ਼ຯͷग़͍ͯͬͨྔΛՃ ͑Ε͹ྑ͍͚ͩͰ͢ɻ ΋͏Ұ౓ࣜมܗΛ͢Δ·Ͱ΋ͳ͘ɺ y ํ޲ʹਨ௚ͳ໘͔Βग़͍ͯ͘ਖ਼ຯͷྲྀྔ ࣜ 4.16: y ํ޲ʹਨ௚ͳ໘͔Βग़͍ͯ͘ਖ਼ຯͷྔ dvz dz dxdydz z ํ޲ʹਨ௚ͳ໘͔Βग़͍ͯ͘ਖ਼ຯͷྲྀྔ ࣜ 4.16: z ํ޲ʹਨ௚ͳ໘͔Βग़͍ͯ͘ਖ਼ຯͷྔ dvy dy dxdydz ͜ΕΒΛ଍͢ͱɺ ࣜ 4.16: x, y, z ํ޲ʹਨ௚ͳ໘͔Βग़͍ͯ͘ਖ਼ຯͷྔ dvx dx + dvy dy + dvz dz dxdydz ͱͳΓ·͢ɻ – 29 –
  37. ୈ 4 ষ ൃࢄ ࠓߟ͍͑ͯΔ௚ํମ͸೚ҙͳେ͖͞ʹ͍ͯ͠·ͨ͠ʢͱΓ͋͑ͣͱͯ΋খ͍͞ͱߟ ͍͑ͯ·ͨ͠ʣͷͰɺ୯Ґମੵ͋ͨΓʹग़͍ͯͬͨਖ਼ຯͷྲྀྔ͸ɺ ࣜ 4.16: x, y,

    z ํ޲ʹਨ௚ͳ໘͔Βग़͍ͯ͘ਖ਼ຯͷྔ dvx dx + dvy dy + dvz dz = ∇ · v = div v ͱͳΓ·͢ɻ͜ΕͰൃࢄΛ਺ࣜͰදݱ͢Δ͜ͱ͕Ͱ͖·ͨ͠ɻ Α͘ݟΔͱ ∇ ԋࢉࢠͱͷ಺ੵʹͳ͍ͬͯΔͷ͕Θ͔Γ·͢Ͷɻ – 30 –
  38. ୈ 5 ষ ճస ຊهࣄͰ͸φϒϥԋࢉࢠΛ࢖ͬͨͱͯ΋ॏཁͳʮޯ഑ gradʯ ʮൃࢄ divʯ ʮճస rotʯʹ͍ͭͯͷղઆΛߦ͍·͢ɻॳֶऀ͕ҎԼΛݟͯʮԿͩ͜Ε͸ʁʯͱͳΒͳ͍

    ͨΊͷষͰ͢ɻ* ޯ഑ɿgrad f(∇ f) * ൃࢄɿdiv v or ∇ · v * ճసɿrot v or ∇ × v ͜ΜͳํΛର৅ʹ͍ͯ͠·͢ɻ • rot ʹۤखҙ͕ࣝ͋Δํ • rot ͷҙຯ΋ؚΊͯཧղ͍ͨ͠ํ ͜ΜͳํΛର৅ʹ͍ͯ͠·͢ɻ • grad, div, rot ʹۤखҙ͕ࣝ͋Δํ • grad, div, rot ͷҙຯ΋ؚΊͯཧղ͍ͨ͠ํ https://takun-physics.net/4487/ 5.1 ճసɿrotation ਺ֶه߸Ͱͷճసͷදݱ͸$\mathrm{rot}$͔ ∇× ͱͳΓ·͢ɻ ∇ ԋࢉࢠͱ͋ΔϕΫτϧ৔ v ͱͷ֎ੵͰ͋Δͱ͓͖֮͑ͯ·͠ΐ͏ɻ @href{https://takun-physics.net/11678/} ͳͥ rot ͱॻ͘ͷ͔ͱ͍͏ͱճసͷӳޠ͕ rotation ͔ͩΒͦͷ๯಄ 3 จࣈΛ࢖ͬ ͯ rot ͱॻ͘ͷͰ͢ɻ ·ͨɺ∇ = ( ∂ ∂x , ∂ ∂y , ∂ ∂z ) Λ࢖ͬͯɺ͋ΔϕΫτϧͱͷ֎ੵΛͱΔͱͦΕ͸ճసΛҙ ຯ͢Δ͜ͱʹͳΓ·͢ɻ ࣜ 5.11: rot rot v = ∇ × v ͱॻ͖·͢ɻ ͜ΕΛ੒෼͝ͱʹॻ͍ͯΈ·͢ɻ – 31 –
  39. ୈ 5 ষ ճస ࣜ 5.11: rot ͷܭࢉ  

    ∂ ∂x ∂ ∂y ∂ ∂z   ×   vx vy vz   =    ∂vz ∂y − ∂vy ∂z ∂vx ∂z − ∂vz ∂x ∂vy ∂x − ∂vx ∂y    ͜ͷΑ͏ʹͳΓ·͢ɻ ݟͯͷ௨Γ grad ΍ div ͱൺֱ͢ΔͱΊͪΌ֮͑ʹ͍͘Ͱ͢ɻ Ͱ͢ͷͰ֮͑ํΛࣗ෼ͳΓʹ͓࣋ͬͯ͘ඞཁ͕͋Γ·͢ɻ ͜͜Ͱ͸ɺ2 ௨Γͷ֮͑ํΛ঺հ͠·͢ͷͰ֮͑΍͍͢ํΛࣗ෼ͳΓʹબΜͰ֮͑ ͓͖ͯ·͠ΐ͏ɻ 3 ߦ 3 ྻͷߦྻ͔ࣜΒʮαϥεͷެࣜʯΛ࢖͏ ΋͠ઢܗ୅਺ΛطʹཤमࡁΈͰʮαϥεͷެࣜΛ஌͍ͬͯΔΑʯͬͯํͰͨ͠Β֮ ͑΍͍͢ํ๏ͩͱࢥ͍·͢ɻ ࣜ 5.11: i j k ∂ ∂x ∂ ∂y ∂ ∂z vx vy vz ͜ͷΑ͏ʹ 3 ߦ 3 ྻͷߦྻࣜΛ༻ҙ͠·͢ɻ ͜͜Ͱɺi, j, k ͸ͦΕͧΕ x, y, z ํ޲ͷ୯ҐϕΫτϧΛද͍ͯ͠·͢ɻ ʮαϥεͷ ެࣜʯ͸ 3 ߦ 3 ྻͷߦྻࣜΛܭࢉ͢Δͱ͖ʹͷެࣜͰ͕͢ɺެࣜ௨Γʹै͑͹ ∇ ԋ ࢉࢠͷ֎ੵΛܭࢉ͍ͯ͠Δ͜ͱʹͳΓ·͢ɻ Ͱ͸ɺ ʮαϥεͷެࣜʯ௨Γʹܭࢉͯ͠Έ·͢ɻ ˛ ਤ 5.1: αϥεͷެࣜ – 32 –
  40. 5.1 ճసɿrotation (ᶃʴᶄʴᶅ)-(ᶆ + ᶇ + ᶈ) Λܭࢉ͢Ε͹ྑ͍Ͱ͢ɻ ࣜ 5.4:

    ∂vz ∂y i + ∂vx ∂z j + ∂vy ∂x k − ∂vx ∂y k + ∂vy ∂z i + ∂vz ∂x j ͜ΕΛ΋͏গ͠·ͱΊΔͱɺ ∂vz ∂y − ∂vy ∂z i + ∂vx ∂z − ∂vz ∂x j + ∂vy ∂x − ∂vx ∂y k ͱɺ͜ͷΑ͏ʹͳΓ·͢ɻ ֤੒෼ΛݟΔͱ͔֬ʹ ∇ ԋࢉࢠͱͷ֎ੵͷ݁Ռͱಉ͡Ͱ͢Ͷɻ ·ͨผͷํ๏ͱͯ͠ ∇ ԋࢉࢠͱͷ֎ੵΛ͍֮͑ͯ·͢ɻ∇ ԋࢉࢠͱͷ֎ੵͱ͍͏ Θ͚Ͱ͸ͳ͘ɺ֎ੵͷܭࢉͦͷ΋ͷͰ͋ΔͨΊɺϕΫτϧͷ֎ੵܭࢉʹ΋࢖͏͜ͱ͕ Ͱ͖·͢ɻ ʮΫϩεͯ͠ʯ ʮͦΕҎ֎ͷ੒෼ʹॻ͘ʯ ʮΫϩεͯ͠ʯ ʮͦΕҎ֎ͷ੒෼ʹॻ͘ʯ ɾɾɾͱढจͷΑ͏ʹ͍֮͑ͯΔ͜ͷํ๏ ͸ɺ݁ߏྑ͍͔ͳͱࢥ͍ͬͯ·͢ɻ ʮ୭ʹशͬͨͷ͔͸๨Ε·͕ͨ͠ɾɾɾʯ खॱ͸͜Μͳײ͡Ͱ͢ɻ ˛ ਤ 5.2: ϕΫτϧͷ֎ੵʢz ੒෼ʣ – 33 –
  41. ୈ 5 ষ ճస ˛ ਤ 5.3: ϕΫτϧͷ֎ੵʢy ੒෼ʣ ˛

    ਤ 5.4: ϕΫτϧͷ֎ੵʢx ੒෼ʣ ݴ༿Ͱॻ͘ΑΓ΋ֆΛ࢖ͬͨํ͕Θ͔Γ΍͍͢ͷͰֆͰදݱͯ͠Έ·ͨ͠ɻ ʮͲ͏Ͱ͔͢Ͷɻ݁ߏ֮͑΍͍͢ͱࢥ͏ͷͰ͕͢Ͷɻʯ 5.1.1 ͳͥ ∇ ԋࢉࢠͱͷ֎ੵ͕ճసͳͷ͔Λಋग़͢Δ ͯ͞ɺnabla ԋࢉࢠͷ֎ੵͷܗࣜ͸Θ͔Γ·͕ͨ͠ɺճస rot ͱදݱ͞ΕΔॴҎΛ ཧղ͢Δඞཁ͕͋Γ·͢ɻ – 34 –
  42. 5.1 ճసɿrotation ൃࢄͷ৔߹͸ʮ͋ΔඍখମੵΛग़ೖΓͨ͠ྔʯͰͨ͠ΑͶɻ ճసͷ৔߹͸ʮ͋ΔඍখྖҬ·ΘΓͷϞʔϝϯτʯΛߟ͑Ε͹ྑ͍ͷͰ͢ɻϞʔϝ ϯτͳͷͰճసͬͯ͜ͱͰ͢ɻ ˛ ਤ 5.5: ఺ (x,

    y) ·ΘΓͷϞʔϝϯτ ؆୯ͷͨΊʹ 2 ࣍ݩʹͯ͠ɺਤ 5.5 ͷΑ͏ʹ͋Δ఺ (x, y) ·ΘΓͷϞʔϝϯτΛߟ ͑ͯΈ·͠ΐ͏ɻ ˞ 2dx = 2dy = 2da ͷඍখྖҬͰͷ z ࣠ํ޲·ΘΓͷϞʔϝϯτΛߟ͑·͢ɻ ൓࣌ܭճΓΛਖ਼ํ޲ʹͯ͠ɾɾɾɾ ࣜ 5.11: vy (x + dx, y)dx + vx (x, y − dy)dy − vx (x, y + dy)dy − vy (x − dx, y)dx ࣜ 5.11: vy (x + dx, y) − vy (x − dx, y) 2dx 2dx dy − vx (x, y + dy) − vx (x, y − dy) 2dy 2dx dy – 35 –
  43. ୈ 5 ষ ճస ࣜ 5.11: ∂vy ∂x − ∂vx

    ∂y 2da2 ͱͳΓ·͢ɻ ࣜ 5.11: 1 2 ∂vy ∂x − ∂vx ∂y 4da2 ͱ͓ͯ͘͠ͱɺඍখྖҬ 4dx dy = 4da2 ͸೚ҙʹબΜͩྖҬͰ͢ͷͰɺ୯Ґ໘ੵ͋ ͨΓͷϞʔϝϯτ͸ɺ ࣜ 5.11: ୯Ґ໘ੵ͋ͨΓͷϞʔϝϯτ 1 2 ∂vy ∂x − ∂vx ∂y ͱͳΓ·͢ɻ1 2 ͕͍ͭͯ͠·͍ͬͯ·͕͢ɺຊ࣭తʹ͸͜Ε͸ z ࣠·ΘΓͷճసΛ ҙຯ͍ͯ͠·͢ɻ z ࣠·ΘΓͷճస͸ɺ ࣜ 5.11: z ࣠·ΘΓͷճస ∇ × v z = ∂vy ∂x − ∂vx ∂y ͱͳΔͷͰɺrot ͷ݁Ռ͸ z ࣠·ΘΓͷճసʹ૬౰͢ΔྔͰ͋Δ͜ͱ͕Θ͔Γ·͢ɻ ͜ΕΛ x ࣠·ΘΓͱ y ࣠·ΘΓϞʔϝϯτʢճసʣΛߟ͑Δͱɺ3 ࣠ͰͷճసΛߟ ͑Δ͜ͱ͕Ͱ͖Δͱ͍͏Θ͚Ͱ͢ɻ – 36 –
  44. None
  45. ϕΫτϧղੳೖ໳ ॳֶऀͷͨΊޯ഑ grad, ൃࢄ div, ճస rot 2021 ೥ 5

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