1. notice
  3. English
  4. 1. feature
  6. logic-topic
  7. 2. logic
  8. 3. set-theory
  9. 4. map
  10. 5. order
  11. 6. combinatorics
  13. calculus
  14. 7. real-numbers
  15. 8. limit-sequence
  16. 9. โ„^n
  17. 10. Euclidean-space
  18. 11. Minkowski-space
  19. 12. polynomial
  20. 13. analytic-Euclidean
  21. 14. analytic-Minkowski
  22. 15. analytic-struct-operation
  23. 16. ordinary-differential-equation
  24. 17. volume
  25. 18. integral
  26. 19. divergence
  27. 20. limit-net
  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-action
  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
  69. 56. feature
  71. ้€ป่พ‘
  72. 57. ้€ป่พ‘
  73. 58. ้›†ๅˆ่ฎบ
  74. 59. ๆ˜ ๅฐ„
  75. 60. ๅบ
  76. 61. ็ป„ๅˆ
  78. ๅพฎ็งฏๅˆ†
  79. 62. ๅฎžๆ•ฐ
  80. 63. ๆ•ฐๅˆ—ๆž้™
  81. 64. โ„^n
  82. 65. Euclidean ็ฉบ้—ด
  83. 66. Minkowski ็ฉบ้—ด
  84. 67. ๅคš้กนๅผ
  85. 68. ่งฃๆž (Euclidean)
  86. 69. ่งฃๆž (Minkowski)
  87. 70. ่งฃๆž struct ็š„ๆ“ไฝœ
  88. 71. ๅธธๅพฎๅˆ†ๆ–น็จ‹
  89. 72. ไฝ“็งฏ
  90. 73. ็งฏๅˆ†
  91. 74. ๆ•ฃๅบฆ
  92. 75. ็ฝ‘ๆž้™
  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. ็พคไฝœ็”จ
  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 . ๅฏไปฅ็†่งฃไธบ็พคไฝœ็”จ็š„ orbit 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 , ๆปก่ถณ

ๅœจๅๆ ‡ไธญ

็š„ metric dual ๅฐฑๆ˜ฏ metric dual ็š„้€†

ๅ› ๆญคไนŸๆœ‰ๅ‡ๆŒ‡ๆ ‡

ไนŸๅฏไปฅๅ่ฟ‡ๆฅ, ไปŽ ็š„้€† ๅ‡บๅ‘, ่ฏๆ˜Žๅฎƒๆ˜ฏ ็š„ๆŸไธช metric ็š„ metric dual, ไธ”่ฟ™ไธช metric ๅœจ็Ÿฉ้˜ต่กจ็คบไธ‹ๆ˜ฏ

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

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

ๅฏผๅ‡บๅŸบ ไธŽๅฏผๅ‡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 ‡