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review derivation: Euler equation proof

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
10 make expr power
  1. 4923339482; locally 3784785:
    \(i x = \log(y)\)
    \(pdg_{1464} pdg_{4621} = \frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}}\)
  1. 0006656532:
    \(e\)
    \(pdg_{2718}\)
  1. 9482923849; locally 9587572:
    \(\exp(i x) = y\)
    \(e^{pdg_{1464} pdg_{4621}} = pdg_{1452}\)
LHS diff is -pdg2718**(pdg1464*pdg4621) + exp(pdg1464*pdg4621) RHS diff is pdg1452 - pdg2718**(log(pdg1452)/log(10)) 4923339482:
9482923849: error for dim with 9482923849
 4923339482:
9482923849: N/A
7 indefinite integrate RHS over
  1. 9482113948; locally 8883737:
    \(\frac{dy}{y} = i dx\)
    \(pdg_{4621}\)
  1. 0004264724:
    \(y\)
    \(pdg_{1452}\)
  1. 9482943948; locally 9984877:
    \(\log(y) = i dx\)
    \(\frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}} = pdg_{4621} pdg_{9199}\)
Nothing to split 9482113948:
9482943948:
9482113948:
9482943948:
5 multiply both sides by
  1. 9848294829; locally 9038289:
    \(\frac{d}{dx} y = y i\)
    \(\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{1452} pdg_{4621}\)
  1. 0003954314:
    \(dx\)
    \(pdg_{9199}\)
  1. 9848292229; locally 1111289:
    \(dy = y i dx\)
    \(pdg_{5842} = pdg_{1452} pdg_{4621} pdg_{9199}\)
LHS diff is -pdg5842 RHS diff is 0 9848294829:
9848292229:
9848294829:
9848292229:
3 factor out X from RHS
  1. 9429829482; locally 1838300:
    \(\frac{d}{dx} y = -\sin(x) + i\cos(x)\)
    \(\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{4621} \cos{\left(pdg_{1464} \right)} - \sin{\left(pdg_{1464} \right)}\)
  1. 0007563791:
    \(i\)
    \(pdg_{4621}\)
  1. 9482984922; locally 2948271:
    \(\frac{d}{dx} y = (i\sin(x) + \cos(x)) i\)
    \(\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{4621} \left(pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\right)\)
LHS diff is 0 RHS diff is (-pdg4621**2 - 1)*sin(pdg1464) 9429829482:
9482984922:
9429829482:
9482984922:
8 indefinite integrate RHS over
  1. 9482943948; locally 9984877:
    \(\log(y) = i dx\)
    \(\frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}} = pdg_{4621} pdg_{9199}\)
  1. 0006563727:
    \(x\)
    \(pdg_{1464}\)
  1. 4928239482; locally 3747585:
    \(\log(y) = i x\)
    \(\frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}} = pdg_{1464} pdg_{4621}\)
no check performed 9482943948:
4928239482:
9482943948:
4928239482:
2 differentiate with respect to
  1. 9492920340; locally 1029383:
    \(y = \cos(x)+i \sin(x)\)
    \(pdg_{1452} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
  1. 0007636749:
    \(x\)
    \(pdg_{1464}\)
  1. 9429829482; locally 1838300:
    \(\frac{d}{dx} y = -\sin(x) + i\cos(x)\)
    \(\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{4621} \cos{\left(pdg_{1464} \right)} - \sin{\left(pdg_{1464} \right)}\)
no check performed 9492920340: error for dim with 9492920340
9429829482:
9492920340: N/A
9429829482:
6 divide both sides by
  1. 9848292229; locally 1111289:
    \(dy = y i dx\)
    \(pdg_{5842} = pdg_{1452} pdg_{4621} pdg_{9199}\)
  1. 0009877781:
    \(y\)
    \(pdg_{1452}\)
  1. 9482113948; locally 8883737:
    \(\frac{dy}{y} = i dx\)
    \(pdg_{4621}\)
Nothing to split 9848292229:
9482113948:
9848292229:
9482113948:
1 declare initial expr
  1. 9492920340; locally 1029383:
    \(y = \cos(x)+i \sin(x)\)
    \(pdg_{1452} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
no validation is available for declarations 9492920340: error for dim with 9492920340
 9492920340: N/A
4 substitute RHS of expr 1 into expr 2
  1. 9492920340; locally 1029383:
    \(y = \cos(x)+i \sin(x)\)
    \(pdg_{1452} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
  2. 9482984922; locally 2948271:
    \(\frac{d}{dx} y = (i\sin(x) + \cos(x)) i\)
    \(\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{4621} \left(pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\right)\)
  1. 9848294829; locally 9038289:
    \(\frac{d}{dx} y = y i\)
    \(\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{1452} pdg_{4621}\)
valid 9492920340: error for dim with 9492920340
9482984922:
9848294829:
9492920340: N/A
9482984922:
9848294829:
12 declare final expr
  1. 4938429483; locally 8888888:
    \(\exp(i x) = \cos(x)+i \sin(x)\)
    \(e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
no validation is available for declarations 4938429483: error for dim with 4938429483
 4938429483: N/A
9 swap LHS with RHS
  1. 4928239482; locally 3747585:
    \(\log(y) = i x\)
    \(\frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}} = pdg_{1464} pdg_{4621}\)
  1. 4923339482; locally 3784785:
    \(i x = \log(y)\)
    \(pdg_{1464} pdg_{4621} = \frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}}\)
valid 4928239482:
4923339482:
4928239482:
4923339482:
11 substitute RHS of expr 1 into expr 2
  1. 9482923849; locally 9587572:
    \(\exp(i x) = y\)
    \(e^{pdg_{1464} pdg_{4621}} = pdg_{1452}\)
  2. 9492920340; locally 1029383:
    \(y = \cos(x)+i \sin(x)\)
    \(pdg_{1452} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
  1. 4938429483; locally 8888888:
    \(\exp(i x) = \cos(x)+i \sin(x)\)
    \(e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\)
valid 9482923849: error for dim with 9482923849
9492920340: error for dim with 9492920340
4938429483: error for dim with 4938429483
 9482923849: N/A
9492920340: N/A
4938429483: N/A
Physics Derivation Graph: Steps for Euler equation proof

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
1464 variable x
\(x\)		 ['real'] dimensionless 140
1452 variable y
\(y\)		 ['real'] dimensionless 13
4621 variable i
\(i\)		 ['imaginary'] dimensionless imaginary unit 74
5842 variable dy
\(dy\)		 ['real']
  • length: 1
 differential displacement along y axis
  • str_note
2
9199 variable dx
\(dx\)		 ['real']
  • length: 1
 15
2718 constant \exp
\(\exp\)		 ['real'] dimensionless e 2.71828   unitless
 8
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