1. notice
  3. ไธญๆ–‡
  4. 1. feature
  6. ้€ป่พ‘
  7. 2. ้€ป่พ‘
  8. 3. ้›†ๅˆ่ฎบ
  9. 4. ๆ˜ ๅฐ„
  10. 5. ๅบ
  11. 6. ็ป„ๅˆ
  13. ๅพฎ็งฏๅˆ†
  14. 7. ๅฎžๆ•ฐ
  15. 8. ๆ•ฐๅˆ—ๆž้™
  16. 9. โ„^n
  17. 10. Euclidean ็ฉบ้—ด
  18. 11. Minkowski ็ฉบ้—ด
  19. 12. ๅคš้กนๅผ
  20. 13. ่งฃๆž (Euclidean)
  21. 14. ่งฃๆž (Minkowski)
  22. 15. ่งฃๆž struct ็š„ๆ“ไฝœ
  23. 16. ๅธธๅพฎๅˆ†ๆ–น็จ‹
  24. 17. ไฝ“็งฏ
  25. 18. ็งฏๅˆ†
  26. 19. ๆ•ฃๅบฆ
  27. 20. ็ฝ‘ๆž้™
  28. 21. ็ดง่‡ด
  29. 22. ่ฟž้€š
  30. 23. ๆ‹“ๆ‰‘ struct ็š„ๆ“ไฝœ
  31. 24. ๆŒ‡ๆ•ฐๅ‡ฝๆ•ฐ
  32. 25. ่ง’ๅบฆ
  34. ๅ‡ ไฝ•
  35. 26. ๆตๅฝข
  36. 27. ๅบฆ่ง„
  37. 28. ๅบฆ่ง„็š„่”็ปœ
  38. 29. Levi-Civita ๅฏผๆ•ฐ
  39. 30. ๅบฆ่ง„็š„ๆ›ฒ็Ž‡
  40. 31. Einstein ๅบฆ่ง„
  41. 32. ๅธธๆˆช้ขๆ›ฒ็Ž‡
  42. 33. simple-symmetric-space
  43. 34. ไธปไธ›
  44. 35. ็พคไฝœ็”จ
  45. 36. ็ƒๆžๆŠ•ๅฝฑ
  46. 37. Hopf ไธ›
  48. ๅœบ่ฎบ
  49. 38. ้ž็›ธๅฏน่ฎบ็‚น็ฒ’ๅญ
  50. 39. ็›ธๅฏน่ฎบ็‚น็ฒ’ๅญ
  51. 40. ็บฏ้‡ๅœบ
  52. 41. ็บฏ้‡ๅœบ็š„ๅฎˆๆ’ๆต
  53. 42. ้ž็›ธๅฏน่ฎบ็บฏ้‡ๅœบ
  54. 43. ๅ…‰้”ฅๅฐ„ๅฝฑ
  55. 44. ๆ—ถ็ฉบๅŠจ้‡็š„่‡ชๆ—‹่กจ็คบ
  56. 45. Lorentz ็พค
  57. 46. ๆ—‹้‡ๅœบ
  58. 47. ๆ—‹้‡ๅœบ็š„ๅฎˆๆ’ๆต
  59. 48. ็”ต็ฃๅœบ
  60. 49. ๅผ ้‡ๅœบ็š„ Laplacian
  61. 50. Einstein ๅบฆ่ง„
  62. 51. ็›ธไบ’ไฝœ็”จ
  63. 52. ่ฐๆŒฏๅญ้‡ๅญๅŒ–
  64. 53. ๆ—‹้‡ๅœบๆ‚้กน
  65. 54. ๅ‚่€ƒ
  67. English
  68. 55. notice
  69. 56. feature
  71. logic-topic
  72. 57. logic
  73. 58. set-theory
  74. 59. map
  75. 60. order
  76. 61. combinatorics
  78. calculus
  79. 62. real-numbers
  80. 63. limit-sequence
  81. 64. โ„^n
  82. 65. Euclidean-space
  83. 66. Minkowski-space
  84. 67. polynomial
  85. 68. analytic-Euclidean
  86. 69. analytic-Minkowski
  87. 70. analytic-struct-operation
  88. 71. ordinary-differential-equation
  89. 72. volume
  90. 73. integral
  91. 74. divergence
  92. 75. limit-net
  93. 76. compact
  94. 77. connected
  95. 78. topology-struct-operation
  96. 79. exponential
  97. 80. angle
  99. geometry
  100. 81. manifold
  101. 82. metric
  102. 83. metric-connection
  103. 84. geodesic-derivative
  104. 85. curvature-of-metric
  105. 86. Einstein-metric
  106. 87. constant-sectional-curvature
  107. 88. simple-symmetric-space
  108. 89. principal-bundle
  109. 90. group-action
  110. 91. stereographic-projection
  111. 92. Hopf-bundle
  113. field-theory
  114. 93. point-particle-non-relativity
  115. 94. point-particle-relativity
  116. 95. scalar-field
  117. 96. scalar-field-current
  118. 97. scalar-field-non-relativity
  119. 98. projective-lightcone
  120. 99. spacetime-momentum-spinor-representation
  121. 100. Lorentz-group
  122. 101. spinor-field
  123. 102. spinor-field-current
  124. 103. electromagnetic-field
  125. 104. Laplacian-of-tensor-field
  126. 105. Einstein-metric
  127. 106. interaction
  128. 107. harmonic-oscillator-quantization
  129. 108. spinor-field-misc
  130. 109. reference

note-math

[orientation-of-real-linear-space] ๆ–นๅ‘

ๆœ‰ไธคไธชๆ–นๅ‘. ๅฏนไบŽ ๅ‘้‡ๅŸบ, ไบคๆขไธ€ๆฌก้กบๅบ ไผšไฝฟๅพ—ๆ–นๅ‘ๆ”นๅ˜, ๅผ•ๅ…ฅ ๅ› ๅญ. ่ฟ™ๅ’Œไบค้”™ๅผ ้‡ๆœ‰็›ธไผผไน‹ๅค„. ๆ–นๅ‘ๅฎšไน‰ไธบๅŸบ็š„ๅŒๅ‘ quotient, ็ญ‰ไปทไบŽ ็š„ ๅˆ†่งฃ

[orientation-of-boundary-of-simplex]

simplx ๆœ‰ๅ‘่พน็•Œ. simplex ่พน็•Œ ็š„ๆ–นๅ‘ๆ˜ฏ, ็ป™่พน็•Œๆ‰€ๅœจ็š„ ไปฟๅฐ„ๅญ็ฉบ้—ดๅฎšไน‰ๆ–นๅ‘, ไฝฟๅพ—ๅ†…้ƒจ ๅœจ ็ปดๆญฃๆ–นๅ‘, ๅค–้ƒจ ๅœจ ็ปด่ดŸๆ–นๅ‘

ๅฆ‚ๆžœๅฏน่พน็•Œ็š„่พน็•Œ็ปง็ปญๅฎšไน‰ๆ–นๅ‘, ๅฐฑไผšๅ‘็Žฐ็›ธ้‚ปๆ–นๅ‘ๆŠตๆถˆ

simplex ้กถ็‚นๆ นๆฎ ๅฏไปฅๆž„้€  ๆœ‰ๅ‘ๅŸบ. ็ฝฎๆขไฝฟๅพ—ๆ–นๅ‘็›ธๅทฎ

้€‰ๅ– ไธบ ๆญฃๆ–นๅ‘ๅŽ, ่พน็•Œ็š„ๆ–นๅ‘ๆ˜ฏ

ๅฏน box ไนŸ็ฑปไผผ

Example ๅ››้ขไฝ“, ๅณๆ‰‹ๅฎšๅˆ™, ๆ‹‡ๆŒ‡ๆŒ‡ๅ‘ๅ››้ขไฝ“ๅ†…้ƒจๅพ—ๅˆฐ่พน็•Œๆ–นๅ‘ (ๅ›พ็‰‡็š„้กถ็‚น็š„ๆŒ‡ๆ ‡ไปŽ ่€Œไธๆ˜ฏ ๅผ€ๅง‹)

[orientable-low-dim-polyhera] ๅคš้ขไฝ“ ๅฏๅฎšๅ‘ ๅฎšไน‰ไธบ, ็”จ simplex ๆž„้€ ๅคš้ขไฝ“ๆ—ถ, ่ƒฝๅคŸๅฏนๆ‰€ๆœ‰ simplex ๅฎšไน‰ๅ…ผๅฎน็š„ๆ–นๅ‘, ไฝฟๅพ—็›ธ้‚ปไธคไธช simplex ็š„ ็›ธๆŽฅ่พน็•Œ simplex ็š„ๆ–นๅ‘ๅ…ผๅฎน i.e. ๆ–นๅ‘ ๅฏนๅบ” simplex ็š„ๅ†…้ƒจๅ’Œ simplex ็š„ๅค–้ƒจ. ๆ–นๅ‘ ๅฏนๅบ” simplex ็š„ๅ†…้ƒจๅ’Œ simplex ็š„ๅค–้ƒจ. i.e. simplex ๅˆ†ๅ‰ฒๆœ‰่‰ฏๅฎšไน‰ๅ†…้ƒจๅ’Œๅค–้ƒจ

Eaxmple [Mobius-strip] ไธๅฏๅฎšๅ‘็š„ Mobius ๅž‹ๅคš้ขไฝ“ (image modified from wiki)

ไธ็ฎกๆ€Žไนˆๅฎšไน‰ๆฏไธช simplex ็š„ๆ–นๅ‘, ้ƒฝๅญ˜ๅœจไธ€ๅฏน็›ธ้‚ป simplex ็š„ ็›ธๆŽฅ่พน็•Œ simplex ็š„ๆ–นๅ‘ไธๅ…ผๅฎน. i.e. simplex ๅˆ†ๅ‰ฒๆฒกๆœ‰่‰ฏๅฎšไน‰็š„ๅ†…้ƒจๅ’Œๅค–้ƒจ

ไปŽๅˆๅง‹ simplex ๅผ€ๅง‹, ไธๆ–ญไผ ้€’ๅœฐๅฏน็›ธ้‚ป simplex ๅฎšไน‰ๅ…ผๅฎน็š„ๆ–นๅ‘, ็ป•ไธ€ๅœˆไผšๅฏผ่‡ด็›ธๆŽฅ่พน็•Œ simplex ็š„ๆ–นๅ‘ไธๅ…ผๅฎน. ๆ–นๅ‘ ้ƒฝๅฏนๅบ” ๅ†…้ƒจ, ๆ–นๅ‘ ้ƒฝๅฏนๅบ” ๅค–้ƒจ

[simplex-chain] simplex chain

[boundary-operator]

่พน็•Œ็ฎ—ๅญ

boundary

Example

  • boundary-op-not-injective

  • [tri-intersect-boundary]

cycle

or

or

[simplex-homology]

k-th homology

where ๅœจ chain ็ฉบ้—ด

็”ฑไบŽๅ‡ ไฝ•ๆ„ไน‰, ๅช้œ€่ฆ ็ณปๆ•ฐ

[real-linear-space-trivial-homology]

is trivial homology or or in , ็š„่พน็•Œๆ˜ฏ้›ถ <==> ๆ˜ฏ่พน็•Œ

Try to prove it by purely affine orientation & combinatorics technique, avoid Euclidean topology

[existence-and-uniqueness-of-n-simplex-chain-with-boundary]

in , uniqueness chain of boundary

so existence of boundary of nonzero chain

and uniqueness of dim region surround by boundary

[homology-hole] ๅฏนไบŽ้›†ๅˆ ๅ‡ๅŽปๆœ‰้™ไธชๆˆ–ๅฏๆ•ฐไธชๅˆ†็ฆป็š„็บฟๆ€งๅญ็ฉบ้—ดๆˆ–่€…ๅคš้ขไฝ“, homology ไธๆ˜ฏ้›ถ

[Stokes-theorem]

็ฑปไผผไบŽไธ€็ปด ๅพฎ็งฏๅˆ†ๅŸบๆœฌๅฎš็†. ็›ด่ง‚ไธŠ, ๆ•ฃๅบฆๅ’Œๆ•ฃๅบฆๅฎš็† = ้ซ˜็ปดๅพฎ็งฏๅˆ†ๅŸบๆœฌๅฎš็†

ๅœจๅๆ ‡้‡Œๅฎšไน‰ [exterior-differential] , ๅ…ถไธญ ๆ˜ฏๅๆ ‡็š„ไฝ“็งฏ, ๆ˜ฏไธ€ๅคง็ฑปๅž‹ๅŒบๅŸŸ, ่ฎก็ฎ—็ป“ๆžœไธไพ่ต–ไบŽๅๆ ‡้€‰ๅ–

ๅˆ™ๆœ‰ Stokes-theorem

for ๅฏๅฎšๅ‘ ็š„ๅ‡ ไนŽๅค„ๅค„่งฃๆž็š„ๅธฆ่พนๆตๅฝข, or

ๅๆ ‡ไธญๅˆฉ็”จ box ่ฎก็ฎ— , ๅ…จ้ƒจๅๆ ‡่ถ‹ไบŽ , ๅฐ†ไผšๆ˜ฏๅฏนๆฏไธชๅๆ ‡่ฝดๆ–นๅ‘่ฎก็ฎ—ๅฏนๆŸไบ›ไธœ่ฅฟ็š„ๅๅพฎๅˆ† , ็ป“ๆžœๆ˜ฏ , ่ฟ›ไธ€ๆญฅ็ฎ€ๅŒ–ๆš‚็•ฅ

Question simplex ไธญๅฟƒไปฟๅฐ„ๅๆ ‡ไธ‹, ๅค–ๅพฎๅˆ†็š„่ฎก็ฎ—็ป“ๆžœ็š„ๅฝขๅผๆ˜ฏไป€ไนˆ?

ไฝ†ๆ˜ฏๅœจไธ€็ปดๅพฎ็งฏๅˆ†ๅŸบๆœฌๅฎš็†็š„่ฏๆ˜Žไธญ, ไธ€็ปดๅŒบ้—ด็š„ๅˆ†ๅ‰ฒ, ไธ€็ปดๅŒบ้—ด็š„่พน็•Œ, ไธ€็ปดๅŒบ้—ด็š„่พน็•Œ็š„็งฏๅˆ†, ้ƒฝๅคช็ฎ€ๅ•ไบ†, ้ซ˜็ปดๅŒบๅŸŸๆฒก้‚ฃไนˆ็ฎ€ๅ•

[Stokes-theorem-simple] ๅฏนไบŽ้ซ˜็ปด, ๅฆ‚ๆžœๆ˜ฏๅผฏๆ›ฒ็š„, ๅˆ™ๅพˆๅ›ฐ้šพ. ๅ…ˆๅค„็†็›ด็š„ไธœ่ฅฟ i.e. simplex or ๅนณ่กŒไฝ“. ๅˆ†ๅ‰ฒไนŸๆ˜ฏๅŒ็ฑปๅž‹ๅŒบๅŸŸ, ่พน็•ŒๆŠตๆถˆไนŸๅพˆ็ฎ€ๅ•. ๅ†็ฑปไผผไบŽไธ€็ปด, ็”จๅพฎๅˆ†ไธญๅ€ผๅฎš็†่ฟ‘ไผผ + compact ๆŽงๅˆถๅณๅฏ. ่ฟ™ๆ ทๅฐฑ่ฏๆ˜Žไบ† simplex or ๅนณ่กŒไฝ“็š„ Stokes ๅฎš็†

[Stokes-theorem-proof] Question

ไฝฟ็”จๆตๅฝขไธŠ็š„ๅฏน form ็งฏๅˆ†็š„ๅฎšไน‰ๆ‰€ไฝฟ็”จ็š„่ฟ‘ไผผๆ–นๆณ• integral-on-manfold

่ฟ‘ไผผๅœฐๅˆ†่งฃไธบ simplex or box, ็„ถๅŽ็”จ simplex ็š„ stokes ๅฎš็† + ๅ†…้ƒจ่พน็•ŒๆŠตๆถˆ, ๅฐฑๅชๅ‰ฉไธ‹็œŸๆญฃ็š„ๆตๅฝข็š„่พน็•Œ

้œ€่ฆไฝฟ็”จ form ๅฏนๅญๆตๅฝข็š„็งฏๅˆ† integral-on-submanfold

่พน็•ŒไธŠ็š„้€ผ่ฟ‘ๅฏ่ƒฝ้œ€่ฆ็‰นๅˆซๆณจๆ„. ไพ‹ๅฆ‚, ๅบ”่ฏฅ่ฎฉ่พน็•ŒไธŠ็š„้€ผ่ฟ‘ไฝฟ็”จไธญๅฟƒๅœจ่พน็•ŒไธŠ็š„ simplex (box) ไปฅๅŠๅพฎๅˆ† at ่พน็•ŒไธŠ็š„็‚น

ๅคงๆฆ‚้œ€่ฆ form ็š„ๆŸ็ง Sobolev ๆŽงๅˆถ?

ๅƒ Gaussโ€“Bonnet ๅฎš็† of Euclidean metric manifold ่ฟ™ๆ ท็š„ไธœ่ฅฟๅบ”่ฏฅไนŸๅฏไปฅ็”จ่ฟ™็งๆ–นๆณ•ๆฅ่ฏๆ˜Ž. ่™ฝ็„ถ่ฟ˜้œ€่ฆ็ปง็ปญ่€ƒ่™‘, ไธบไป€ไนˆ็ป“ๆžœๆ˜ฏไธๅ— metric ๅฝฑๅ“็š„ homology invariant Euler characteristic (ๅทฎไธ€ไธช ็ปด Euclidean ็š„ไฝ“็งฏๅ› ๅญ, ไปฅ ็š„ๅน‚่กจ็คบ)

ๆˆ‘ๅนถๆฒกๆœ‰ๅฏนๆฒกๆœ‰่พน็•Œ็š„ๆตๅฝขๅค„็† Stokes ๅฎš็†, ๅนถๆฒกๆœ‰ๅฎšไน‰ . ๆฒกๆœ‰็š„่พน็•Œๆตๅฝข Example

่พน็•Œ็ฎ—ๅญไธŽๅค–ๅพฎๅˆ†็š„ๅฏนๅบ”ๆ€ง่ดจ

homology

cohomology

[coboundary-operator]

coboundary

cocycle . ็›ด่ง‚ๆ˜ฏ่ฟ™ไธ€็‚น็š„ form ็š„ๆ•ฃๅบฆๆ˜ฏ้›ถ

or

or

[de-Rham-cohomolgy] k-th de Rham cohomology

in , cohomology trivial

[cohomology-hole] ๅธฆ "ๆดž" ็š„ form. Example in , or ๅœจ ๆ˜ฏๅฅ‡็‚น. ๅœจ้ž ็š„ๆตๅฝข, ๅฏ่ƒฝๅณไฝฟๅ‡ฝๆ•ฐๆฒกๆœ‰ๅฅ‡็‚น, form ๅ’Œ Stokes ๅฎš็†ไนŸ่ƒฝๅฐ†ๆตๅฝข็š„ๆดž่กจ็Žฐๅ‡บๆฅ. Example or

metric ๆตๅฝข็š„ๆƒ…ๅ†ต

ๅฏน form ็š„็งฏๅˆ†็›ธๅฝ“ไบŽๅฏน ็š„็งฏๅˆ†

[Hodge-star]

Hodge star ็ฎ—ๅญ as form ็š„ๆญฃไบค่กฅๅฏนๅถ

with ==>

==>

[flux]

ๅฏน form ็งฏๅˆ† -> ๅฏน ็งฏๅˆ† -> ๅฏน ็งฏๅˆ†, ่งฃ้‡Šไธบ้€š่ฟ‡ ็š„ๆญฃไบค่กฅ ็š„้‡ ๅฏน ็งฏๅˆ†, i.e. ้€š้‡

็”จ metric-dual ไปฃ่กจ้€š้‡ ไบค้”™ๅผ ้‡, ๅ†…็งฏไปฃ่กจ้‡ ๅœจ้€š้‡ๆ–นๅ‘ ไธŠ็š„ๆญฃไบคๆŠ•ๅฝฑ

Example in Euclidean , (ๅ›พ)

  • form

ๅๆ ‡

Stokes ๅฎš็† [gradient]

  • form

ๆณจๆ„ๆญคๆ—ถๅฏไปฅๅŠ ไธŠๅฏๅฎšๅ‘ไบŒ็ปด็š„ "ๆ—‹่ฝฌ 90 ๅบฆ" ไปŽไบŒ็ปดๆ•ฃๅบฆๅ˜ๆˆไบŒ็ปดๆ—‹ๅบฆ, ๅฏน่พน็•Œ็š„ๆณ•้€š้‡ๅ˜ไธบๅฏน่พน็•Œ็š„ๅˆ‡ๆต้‡

ๅๆ ‡

Stokes ๅฎš็† [curl]

where

  • form

ๅๆ ‡

Stokes ๅฎš็† [divergence]

in Minkowski ,