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