Review Derivation: Physics Derivation Graph

Return to navigation page or list derivations

Review Euler equation proof

abstract:
reference:

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			9492920340 $y = \cos(x)+i \sin(x)$	no validation is available for declarations
2	0000111649: differentiate with respect to number of inputs: 1; feeds: 1; outputs: 1 Differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.	9492920340 $y = \cos(x)+i \sin(x)$	0007636749 $x$	9429829482 $\frac{d}{dx} y = -\sin(x) + i\cos(x)$	valid
3	0000111260: factor out X from RHS number of inputs: 1; feeds: 1; outputs: 1 Factor $#1$ from the RHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	9429829482 $\frac{d}{dx} y = -\sin(x) + i\cos(x)$	0007563791 $i$	9482984922 $\frac{d}{dx} y = (i\sin(x) + \cos(x)) i$	LHS diff is 0 RHS diff is (-pdg0004621*2 - 1)sin(pdg0001464)
4	0000111634: substitute RHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	9482984922 $\frac{d}{dx} y = (i\sin(x) + \cos(x)) i$ 9492920340 $y = \cos(x)+i \sin(x)$		9848294829 $\frac{d}{dx} y = y i$	LHS diff is pdg0001452 RHS diff is -pdg0001452pdg0004621 + pdg0004621sin(pdg0001464) + cos(pdg0001464)
5	0000111182: multiply both sides by number of inputs: 1; feeds: 1; outputs: 1 Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	9848294829 $\frac{d}{dx} y = y i$	0003954314 $dx$	9848292229 $dy = y i dx$	LHS arithmetic error. Diff: -pdg0005842
6	0000111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	9848292229 $dy = y i dx$	0009877781 $y$	9482113948 $\frac{dy}{y} = i dx$	Not evaluated due to missing term in SymPy
7	0000111132: indefinite integrate RHS over number of inputs: 1; feeds: 1; outputs: 1 Indefinite integral of RHS of Eq.~\ref{eq:#2} over $#1$; yields Eq.~\ref{eq:#3}.	9482113948 $\frac{dy}{y} = i dx$	0004264724 $y$	9482943948 $\log(y) = i dx$	recognized infrule but not yet supported
8	0000111132: indefinite integrate RHS over number of inputs: 1; feeds: 1; outputs: 1 Indefinite integral of RHS of Eq.~\ref{eq:#2} over $#1$; yields Eq.~\ref{eq:#3}.	9482943948 $\log(y) = i dx$	0006563727 $x$	4928239482 $\log(y) = i x$	recognized infrule but not yet supported
9	0000111268: swap LHS with RHS number of inputs: 1; feeds: 0; outputs: 1 Swap LHS of Eq.~\ref{eq:#1} with RHS; yields Eq.~\ref{eq:#2}.	4928239482 $\log(y) = i x$		4923339482 $i x = \log(y)$	valid
10	0000111721: make expr power number of inputs: 1; feeds: 1; outputs: 1 Make Eq.~\ref{eq:#2} the power of $#1$; yields Eq.~\ref{eq:#3}.	4923339482 $i x = \log(y)$	0006656532 $e$	9482923849 $\exp(i x) = y$	LHS diff is -pdg0002718*(pdg0001464pdg0004621) + exp(pdg0001464pdg0004621) RHS diff is pdg0001452 - pdg0002718*(log(pdg0001452)/log(10))
11	0000111634: substitute RHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	9492920340 $y = \cos(x)+i \sin(x)$ 9482923849 $\exp(i x) = y$		4938429483 $\exp(i x) = \cos(x)+i \sin(x)$	LHS diff is 0 RHS diff is pdg0001452 - pdg0004621*sin(pdg0001464) - cos(pdg0001464)
12	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	4938429483 $\exp(i x) = \cos(x)+i \sin(x)$			no validation is available for declarations

Symbols used in Euler equation proof

0000002718: $ \exp $
0000009199: $ dx $
0000005842: $ dy $
0000004621: $ i $
0000001464: $ x $
0000001452: $ y $

d3js visualization of steps and expressions in Euler equation proof

Hold the mouse over a node to highlight that node and its neighbors. You can zoom in/out. You can pan the image. You can move nodes by clicking and dragging.

Actions: Edit Derivation

You must be logged in to edit the steps of derivation, edit the abstract, or rename the derivation.

Generate Tex file or PDF file

Delete Derivation and all associated steps

You must be logged in to delete the derivation.

timing of Neo4j queries:

0.005 seconds for pdg_app/to_review_derivation db07d455-9e7f-4220-8584-2562fe57fa89
0.009 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivationd40d9199-a7f9-4b2e-96ba-3ea702e5605c
0.003 seconds for compute/input_feed_output_infrule_for_step: get_inference_rule_connected_to_step_IDd40d9199-a7f9-4b2e-96ba-3ea702e5605c
0.003 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type HAS_INPUTd40d9199-a7f9-4b2e-96ba-3ea702e5605c
0.002 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_FEEDd40d9199-a7f9-4b2e-96ba-3ea702e5605c
0.002 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_OUTPUTd40d9199-a7f9-4b2e-96ba-3ea702e5605c
0.002 seconds for compute/get_dict_of_steps_in_derivation: get_sequence_index_for_stepd40d9199-a7f9-4b2e-96ba-3ea702e5605c
0.004 seconds for pdg_app/ db07d455-9e7f-4220-8584-2562fe57fa89

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			9492920340 \(y = \cos(x)+i \sin(x)\)	no validation is available for declarations
2	0000111649: differentiate with respect to number of inputs: 1; feeds: 1; outputs: 1 Differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.	9492920340 \(y = \cos(x)+i \sin(x)\)	0007636749 \(x\)	9429829482 \(\frac{d}{dx} y = -\sin(x) + i\cos(x)\)	valid
3	0000111260: factor out X from RHS number of inputs: 1; feeds: 1; outputs: 1 Factor $#1$ from the RHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	9429829482 \(\frac{d}{dx} y = -\sin(x) + i\cos(x)\)	0007563791 \(i\)	9482984922 \(\frac{d}{dx} y = (i\sin(x) + \cos(x)) i\)	LHS diff is 0 RHS diff is (-pdg0004621*2 - 1)sin(pdg0001464)
4	0000111634: substitute RHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	9482984922 \(\frac{d}{dx} y = (i\sin(x) + \cos(x)) i\) 9492920340 \(y = \cos(x)+i \sin(x)\)		9848294829 \(\frac{d}{dx} y = y i\)	LHS diff is pdg0001452 RHS diff is -pdg0001452pdg0004621 + pdg0004621sin(pdg0001464) + cos(pdg0001464)
5	0000111182: multiply both sides by number of inputs: 1; feeds: 1; outputs: 1 Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	9848294829 \(\frac{d}{dx} y = y i\)	0003954314 \(dx\)	9848292229 \(dy = y i dx\)	LHS arithmetic error. Diff: -pdg0005842
6	0000111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	9848292229 \(dy = y i dx\)	0009877781 \(y\)	9482113948 \(\frac{dy}{y} = i dx\)	Not evaluated due to missing term in SymPy
7	0000111132: indefinite integrate RHS over number of inputs: 1; feeds: 1; outputs: 1 Indefinite integral of RHS of Eq.~\ref{eq:#2} over $#1$; yields Eq.~\ref{eq:#3}.	9482113948 \(\frac{dy}{y} = i dx\)	0004264724 \(y\)	9482943948 \(\log(y) = i dx\)	recognized infrule but not yet supported
8	0000111132: indefinite integrate RHS over number of inputs: 1; feeds: 1; outputs: 1 Indefinite integral of RHS of Eq.~\ref{eq:#2} over $#1$; yields Eq.~\ref{eq:#3}.	9482943948 \(\log(y) = i dx\)	0006563727 \(x\)	4928239482 \(\log(y) = i x\)	recognized infrule but not yet supported
9	0000111268: swap LHS with RHS number of inputs: 1; feeds: 0; outputs: 1 Swap LHS of Eq.~\ref{eq:#1} with RHS; yields Eq.~\ref{eq:#2}.	4928239482 \(\log(y) = i x\)		4923339482 \(i x = \log(y)\)	valid
10	0000111721: make expr power number of inputs: 1; feeds: 1; outputs: 1 Make Eq.~\ref{eq:#2} the power of $#1$; yields Eq.~\ref{eq:#3}.	4923339482 \(i x = \log(y)\)	0006656532 \(e\)	9482923849 \(\exp(i x) = y\)	LHS diff is -pdg0002718*(pdg0001464pdg0004621) + exp(pdg0001464pdg0004621) RHS diff is pdg0001452 - pdg0002718*(log(pdg0001452)/log(10))
11	0000111634: substitute RHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	9492920340 \(y = \cos(x)+i \sin(x)\) 9482923849 \(\exp(i x) = y\)		4938429483 \(\exp(i x) = \cos(x)+i \sin(x)\)	LHS diff is 0 RHS diff is pdg0001452 - pdg0004621*sin(pdg0001464) - cos(pdg0001464)
12	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	4938429483 \(\exp(i x) = \cos(x)+i \sin(x)\)			no validation is available for declarations