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

The problem of defining the derivative of a vector field in the direction

For a vector field near , in the direction, try to calculate the derivative in coordinates

However, the difference operation is not linearly compatible with coordinate changes of general diffeomorphisms.

But in a metric-manifold, there are special coordinates โ€” geodesic coordinates. The transformation method for different geodesic coordinates at is , which is linear.

[geodesic-derivative] Geodesic derivative alias [Levi-Civita-derivative] Levi-Civita derivative :=

In geodesic coordinates at point , the derivative at point ,

It is also possible to take the derivative of a tensor field . According to the scalar multiplication associated with the tensor structure, calculations can be performed using the product-rule Example

Prop . Proof In geodesic coordinates

Prop or

Prop The covariant derivative is compatible with the metric-dual e.g. since

Other coordinates may be needed to compute geodesic coordinates, and thus other coordinates may also be needed to express the geodesic derivative.

[geodesic-derivative-in-general-coordinate]

Compute geodesic coordinates using general coordinates , then in coordinates , the geodesic derivative is

Using connection transformation

==>

Use . Substitute into the calculation of

The tangent space linearly transforms to by , but keeps in coordinate , but keep in coordinate

Or written as, in general coordinates, geodesic derivative

For coordinate-frame

Is there a more intuitive explanation, rather than directly using the transformation of connection?

If we only consider linear compatibility, then there are many linear connections, and the one that coincides with the geodesic-derivative is the metric-connection

[geodesic-derivative-of-co-vector] Prop For co-vector field

Proof

Question Similar to the case of vector fields. Use the transformation and the product-rule

For co-vector coordinate-frame

[parallel-transport-metric-connection]

Parallel transport as โ€œzero rate of change along the curveโ€ or where

is an ODE

According to calculation (?), covariant derivative can be recovered from parallel transport + difference quotient

[orthonormal-frame]

Parallel transport of metric-connection preserves the metric

Can be used to construct orthonormal frames

It can be proven that a manifold metric uniquely corresponds to an principal-bundle structure on the manifold

But are there more specific and operational calculation results? Regarding calculating orthonormal frames using parallel transport in geodesic coordinates

A canonical orthonormal frame may be used for simplified some of calculations of spinors on curved manifolds, e.g. Pauli-matrix