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

separable ODE in 1 dimension

where , initial value undetermined

Example

  • .
  • .

[exponential-of-vector-field] Question

let open in

vector-field is analytic function

The exponential-of-vector-field generated by the vector field should be invariant

Taylor series of vector field

polynomial like

or with

make it correspond to ODE

and or is local diffeomorphism. This is the relation between ODE and symmetry, i.e. Lie theory

if we also write as , then

Example

compare to the result from separable ODE in 1 dimension

compare , expect with

,

โ€ฆ

or

Example [harmonic-oscillator]

make harmonic-oscillator first order

with the case of trigonometric

so

or written in the form of complex number and exponential

similar for the case of hyperbolic

The characteristic polynomial equation of harmonic-oscillator is or . We are interested in the trigonometric case or , its prototype is or

In the case where the harmonic oscillator is a real value, for the complex exponential form, in order to remain in , when , the coefficients in front of are complex conjugates of each other

  • ,

compare , expect with

โ€ฆ

or

Question

The in should correspond to the case of a dilation vector field

One-parameter homomorphism embedding

is called flow. exp road emission-like coordinates

[vector-field-as-ฮด-diffeomorphism] Near , the vector field is the coordinate of the diffeomorphism group , similar to geodesic-coordinate

ODE

wiki:Cauchy-Kovalevskaya_theorem, the estimation of the radius of convergence uses a special upper bound control method, similar to what is done in inverse-analytic

, ==>

[integral-curve] Picard iteration of ODE solution (wiki) representation or integral curve e.g.

If it is a linear ODE, then (alias Dyson series)

Linear ODE. The solution of a constant coefficient ODE can be written by transforming it into a first-order differential equation system + Jordan normal form

[Lie-bracket] Lie bracket

as generator of conjugate-action of

conjugation action of the group

Differential :=

note that is a different map, if we consider the order of

for ,

[Lie-derivative] Lie derivative alias drag derivative

let generate a one-parameter diffeomorphism

let

Jacobi identity or

Lie derivative can also be defined for tensor fields โ€ฆ

[linear-PDE-integrable-condition] related to the symmetry of the second-order derivative. This condition allows the solution of a first-order linear PDE to come from successive integration curve of ODEs, and the result does not depend on the choice of path