1. notice
  3. English
  5. logic_topic
  6. 1. logic
  7. 2. basic
  8. 3. map
  9. 4. order
  10. 5. combinatorics
  12. calculus
  13. 6. real_numbers
  14. 7. limit_sequence
  15. 8. division_algebra
  16. 9. Euclidean_space
  17. 10. Minkowski_space
  18. 11. polynomial
  19. 12. analytic_Euclidean
  20. 13. analytic_struct_operation
  21. 14. ordinary_differential_equation
  22. 15. convex_hull
  23. 16. volume
  24. 17. integral
  25. 18. divergence
  26. 19. limit_net
  27. 20. topology
  28. 21. compact
  29. 22. connected
  30. 23. topology_struct_operation
  31. 24. exponential
  32. 25. angle
  34. geometry
  35. 26. manifold
  36. 27. metric
  37. 28. metric_connection
  38. 29. geodesic_derivative
  39. 30. curvature_of_metric
  40. 31. Einstein_metric
  41. 32. constant_sectional_curvature
  42. 33. simple_symmetric_space
  43. 34. principal_bundle
  44. 35. group
  45. 36. stereographic_projection
  46. 37. Hopf_bundle
  48. field_theory
  49. 38. point_particle_non_relativity
  50. 39. point_particle_relativity
  51. 40. scalar_field
  52. 41. scalar_field_current
  53. 42. scalar_field_non_relativity
  54. 43. projective_lightcone
  55. 44. spacetime_momentum_spinor_representation
  56. 45. Lorentz_group
  57. 46. spinor_field
  58. 47. spinor_field_current
  59. 48. electromagnetic_field
  60. 49. Laplacian_of_tensor_field
  61. 50. Einstein_metric
  62. 51. interaction
  63. 52. harmonic_oscillator_quantization
  64. 53. spinor_field_misc
  65. 54. reference
  67. ไธญๆ–‡
  68. 55. notice
  70. ้€ป่พ‘
  71. 56. ้€ป่พ‘
  72. 57. ๅŸบ็ก€
  73. 58. ๆ˜ ๅฐ„
  74. 59. ๅบ
  75. 60. ็ป„ๅˆ
  77. ๅพฎ็งฏๅˆ†
  78. 61. ๅฎžๆ•ฐ
  79. 62. ๆ•ฐๅˆ—ๆž้™
  80. 63. ๅฏ้™คไปฃๆ•ฐ
  81. 64. Euclidean ็ฉบ้—ด
  82. 65. Minkowski ็ฉบ้—ด
  83. 66. ๅคš้กนๅผ
  84. 67. ่งฃๆž (Euclidean)
  85. 68. ่งฃๆž struct ็š„ๆ“ไฝœ
  86. 69. ๅธธๅพฎๅˆ†ๆ–น็จ‹
  87. 70. convex_hull
  88. 71. ไฝ“็งฏ
  89. 72. ็งฏๅˆ†
  90. 73. ๆ•ฃๅบฆ
  91. 74. ็ฝ‘ๆž้™
  92. 75. ๆ‹“ๆ‰‘
  93. 76. ็ดง่‡ด
  94. 77. ่ฟž้€š
  95. 78. ๆ‹“ๆ‰‘ struct ็š„ๆ“ไฝœ
  96. 79. ๆŒ‡ๆ•ฐๅ‡ฝๆ•ฐ
  97. 80. ่ง’ๅบฆ
  99. ๅ‡ ไฝ•
  100. 81. ๆตๅฝข
  101. 82. ๅบฆ่ง„
  102. 83. ๅบฆ่ง„็š„่”็ปœ
  103. 84. Levi_Civita ๅฏผๆ•ฐ
  104. 85. ๅบฆ่ง„็š„ๆ›ฒ็Ž‡
  105. 86. Einstein ๅบฆ่ง„
  106. 87. ๅธธๆˆช้ขๆ›ฒ็Ž‡
  107. 88. simple_symmetric_space
  108. 89. ไธปไธ›
  109. 90. ็พค
  110. 91. ็ƒๆžๆŠ•ๅฝฑ
  111. 92. Hopf ไธ›
  113. ๅœบ่ฎบ
  114. 93. ้ž็›ธๅฏน่ฎบ็‚น็ฒ’ๅญ
  115. 94. ็›ธๅฏน่ฎบ็‚น็ฒ’ๅญ
  116. 95. ็บฏ้‡ๅœบ
  117. 96. ็บฏ้‡ๅœบ็š„ๅฎˆๆ’ๆต
  118. 97. ้ž็›ธๅฏน่ฎบ็บฏ้‡ๅœบ
  119. 98. ๅ…‰้”ฅๅฐ„ๅฝฑ
  120. 99. ๆ—ถ็ฉบๅŠจ้‡็š„่‡ชๆ—‹่กจ็คบ
  121. 100. Lorentz ็พค
  122. 101. ๆ—‹้‡ๅœบ
  123. 102. ๆ—‹้‡ๅœบ็š„ๅฎˆๆ’ๆต
  124. 103. ็”ต็ฃๅœบ
  125. 104. ๅผ ้‡ๅœบ็š„ Laplacian
  126. 105. Einstein ๅบฆ่ง„
  127. 106. ็›ธไบ’ไฝœ็”จ
  128. 107. ่ฐๆŒฏๅญ้‡ๅญๅŒ–
  129. 108. ๆ—‹้‡ๅœบๆ‚้กน
  130. 109. ๅ‚่€ƒ

note-math

[quadratic_form] ไบŒๆฌกๅž‹ alias [metric] ๅบฆ่ง„ alias ๅ†…็งฏ [inner_product]

Example ่ท็ฆป

Example spacetime metric

ๅŒ็บฟๆ€งๅ‡ฝๆ•ฐ, ไบŒๆฌกๅž‹

metric ๅธธ็”จ่ฎฐๅท . ๅœจๅๆ ‡ไธญ่ฎฐไธบ

ๅๆ ‡ไธ‹ metric ๅฏไปฅ่กจ็คบไธบ็Ÿฉ้˜ต ๅ’Œ็Ÿฉ้˜ตไน˜ๆณ•

[signature_inertia] signature ๅœจ ไธ‹ไธๅ˜. ๅฏน็งฐ็Ÿฉ้˜ต ็š„็‰นๅพๅ€ผๅ’Œๅฏน่ง’ๅŒ– ๅ…ถไธญ ไธช , ไธช , ไธช .

ไธๅŒ signature ๅฏไปฅ็†่งฃไธบ ็พคไฝœ็”จๅœจไบŒๆฌกๅž‹็ฉบ้—ด็š„ orbit ็š„ๅˆ†็ฑป

[quadratic_form_non_degenerated] ้ž้€€ๅŒ– := signature ไธญ

้€€ๅŒ–็š„ไบŒๆฌกๅž‹ๅฏไปฅ้™ๅˆถๅœจ ๅญ็ฉบ้—ดไธญๅพ—ๅˆฐ้ž้€€ๅŒ–

ไปฅไธ‹็ญ‰ไปท

  • metric ้ž้€€ๅŒ–

  • [quadratic_form_dual] ๆ˜ฏๅŒๅฐ„ alias [musical_isomorphism]

    ็›ธๅฏนไบŽ metric dual ๆ˜ ๅฐ„่ฎฐไธบ , metric dual ็š„้€†ๆ˜ ๅฐ„่ฎฐไธบ

  • ไบŒๆฌกๅž‹็Ÿฉ้˜ตๅฏ้€†

ๅ›บๅฎšไธ€ไธช้ž้€€ๅŒ–ไบŒๆฌกๅž‹ๆ—ถ, ็ป“ๆž„็พค . ไฟๆŒไธคไธชๆ–นๅ‘ๅฐฑๅพ—ๅˆฐ

metric ็Ÿฉ้˜ต ็š„้€† . ๅœจๅๆ ‡ไธญ่ฎฐไธบ ๅ’Œ

==> ๅœจๅๆ ‡ไธญ

let ๅ‘้‡็ฉบ้—ด็š„ๅŸบ ็š„ๅฏนๅถๅŸบ :=

let then

[rasing_and_lowring_index] ๅ‡้™ๆŒ‡ๆ ‡

quadratic-form-dual ๅœจๅๆ ‡ไธญ็š„็Ÿฉ้˜ต่กจ็คบ

ๅ…ถไธญ ๆ˜ฏ้™ๆŒ‡ๆ ‡. ๆˆ–่€…

ๆณจๆ„ metric ็Ÿฉ้˜ตๆ˜ฏๅฏน็งฐ or

ๆ˜ฏ ็š„ๅฏนๅถๅŸบ่กจ็คบ็š„็ณปๆ•ฐ, ๅ› ไธบ , ๆˆ–่€…็”จ ๆ˜ฏ ๅœจๅฏนๅถๅŸบ ็š„็ณปๆ•ฐ

ๅฏนไบŽ metric ็Ÿฉ้˜ต็š„้€†, ๅฎšไน‰ dual ็ฉบ้—ด ็š„ metric ไธบ . ๅฎƒๆปก่ถณไปฅไธ‹ๅผๅญ

ๅผๅญๅœจๅๆ ‡ไธญ่กจ็คบไธบ

็š„ metric dual ๅฐฑๆ˜ฏ metric dual ็š„้€†, ไปŽ่€ŒไนŸๆ˜ฏ

ๅ› ๆญคไนŸๆœ‰ๅ‡ๆŒ‡ๆ ‡

[tensor] ๅคš้‡็บฟๆ€ง (ๅ…ผๅฎน้›†ๅˆ็งฏ็š„ and ้€ป่พ‘) + ๆœ€ๅฐ็‹ฌ็ซ‹ๆ— ๅ…ณ็”Ÿๆˆ (quotient ๆž„้€ )

ๅฏผๅ‡บๅŸบ , ๅฏผๅ‡บๅŸบ็š„็ณปๆ•ฐ

็”ฑๅผ ้‡็š„ๆ€ง่ดจ

[tensor_induced_quadratic_form]

ๅฐ†ๅ‘้‡็ฉบ้—ด็š„ไบŒๆฌกๅž‹ ๅฏผๅ‡บๅˆฐๅผ ้‡็ฉบ้—ด ็š„ไบŒๆฌกๅž‹

้ๅŽ†ๆ‰€ๆœ‰ๆญฃไบคๅŸบ with

ๅพ—ๅˆฐ signature

[rasing_and_lowring_index_tensor] ๅผ ้‡ไนŸๅฏไปฅ metric dual ๅ‡้™ๆŒ‡ๆ ‡

Example ้™ๆŒ‡ๆ ‡

ๅœจ ็š„ๅฏผๅ‡บ metric ็š„ metric dual ๆ˜ฏ . ๅ่ฟ‡ๆฅไนŸไธ€ๆ ท

่ฎฐ ไธบ

่ฎฐไบŒๆฌกๅž‹ไธบ

่€ƒ่™‘ ็š„็ฎ€ๅ•ๅผ ้‡. ๅฏนไบŽ , ๅˆ™ ๅผ ้‡ ็š„ๅฏผๅ‡บ metric ๆ˜ฏ

Prop trace-identity duality

Proof

Question ๆœ‰ๆฒกๆœ‰ๆ›ดๆœฌ่ดจๆ›ดๆ— ๅๆ ‡็š„่ฏๆ˜Žๆ–นๆณ•?

็ฌฌไธ€็ง่ฏๆ˜Žๆ–นๆณ•ๆ˜ฏ, ไฝฟ็”จไธ€ไธช่ง„่ŒƒๆญฃไบคๅŸบ

ๆญคๆ—ถ

ๅช้œ€่ฆ่ฏๆ˜Ž ็š„ๆƒ…ๅ†ต

็ฌฌไบŒ็ง่ฏๆ˜Žๆ˜ฏๅˆฉ็”จ ๅผ ้‡็š„็ป“ๆžœ, ็„ถๅŽ่ฝฌๅ›ž ๅผ ้‡

ๅœจ ๅผ ้‡ไธญ, ไฝœไธบ ็š„ๅฏนๅถ

Prop let , then

Proof ๅ–ไธ€ไธช่ง„่ŒƒๆญฃไบคๅŸบ , ๆญคๆ—ถ

let defined as

Prop

Proof

Prop

Proof ๅช้œ€่ฏๆ˜Ž็ฎ€ๅ•ๅผ ้‡็š„ๆƒ…ๅ†ต

ไฝ†ๆ˜ฏ

ๆ‰€ไปฅ

ๆœ€ๅŽไธ€ๆญฅๅฏไปฅๅ†ๆพ„ๆธ…ไธ€ไธ‹. ๅฏน

Prop

Proof

Prop

Proof ไธค่พนไฝฟ็”จ

Prop ไฝฟ็”จ็Ÿฉ้˜ต ่กจ็คบ

  • ๅฏนๅบ”
  • ๅฏนๅบ”
  • ไปŽ่€Œ ็š„็Ÿฉ้˜ต่กจ็คบๆ˜ฏ

Prop

Prop ๅœจ่ง„่ŒƒๆญฃไบคๅŸบไธญ ไธ” . ๆญคๆ—ถ

Prop

Prop (ๅœจ่ง„่ŒƒๆญฃไบคๅŸบไธญ)

Prop

Proof and

Prop [Killing_form_of_orthogonal_group] (Killing form of up to a constant)

Proof ๆ นๆฎ