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 $\frac{d}{dx} y = -\sin(x) + i\cos(x)$	recognized infrule but not yet supported
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

Generate Tex file or PDF file

Delete Derivation and all associated steps

This does not remove expressions, symbols, and operations.

timing of Neo4j queries:

0.003 seconds for pdg_app/to_review_derivation a3e34deb-3f3b-4cd0-95b4-1c51217141ba
0.004 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivation05df396d-746e-4af5-aa66-8394d0f36160
0.002 seconds for compute/input_feed_output_infrule_for_step: get_inference_rule_connected_to_step_ID05df396d-746e-4af5-aa66-8394d0f36160
0.002 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type HAS_INPUT05df396d-746e-4af5-aa66-8394d0f36160
0.002 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_FEED05df396d-746e-4af5-aa66-8394d0f36160
0.002 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_OUTPUT05df396d-746e-4af5-aa66-8394d0f36160
0.002 seconds for compute/get_dict_of_steps_in_derivation: get_sequence_index_for_step05df396d-746e-4af5-aa66-8394d0f36160
0.003 seconds for pdg_app/ a3e34deb-3f3b-4cd0-95b4-1c51217141ba

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 \(\frac{d}{dx} y = -\sin(x) + i\cos(x)\)	recognized infrule but not yet supported
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