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

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