An Isabelle/HOL Formalisation of Greens Theorem Mohammad Abdulaziz - PowerPoint PPT Presentation

An Isabelle/HOL Formalisation of Green’s Theorem Mohammad Abdulaziz Data61/ANU and Lawrence Paulson Computer Laboratory, University of Cambridge August 25, 2016 Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 1 / 15

Abstract ◮ We formalised a statement of Green’s theorem in Isabelle/HOL ◮ Outline ◮ What is Green’s theorem? ◮ Traditional statement and proof of Green’s theorem ◮ Our statement and proof of Green’s theorem Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 2 / 15

What is Green’s Theorem? Stokes’ Theorems ◮ A family of theorems relating functions to the integrals of their derivatives ◮ 1 dimension: Fundamental Theorem of Calculus, for f : R ⇒ R � b df dx dx = f ( b ) − f ( a ) a ◮ 2 dimension: Green’s Theorem for a field F : R 2 ⇒ R 2 � � ∂ F y ∂ x − ∂ F x F x dx + F y dy = ∂ y dxdy ∂ D D Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 3 / 15

What is Green’s Theorem? Green’s Theorem ◮ a region D : R 2 set : satisfying some conditions D = { ( x , y ) | x 2 + y 2 ≤ C } y x Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 4 / 15

What is Green’s Theorem? Green’s Theorem ◮ a field F : R 2 ⇒ R 2 : satisfying some conditions in and around D F ( x , y ) = ( F x ( x , y ) , F y ( x , y )) F x ( x , y ) = y 3 − 9 y F y ( x , y ) = x 3 − 9 x Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 4 / 15

What is Green’s Theorem? Green’s Theorem y x Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 4 / 15

What is Green’s Theorem? Green’s Theorem ∂ F y ∂ x − ∂ F x � � F x dx + F y dy = ∂ y dxdy ∂ D D y x Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 4 / 15

What is Green’s Theorem? Line integral ∂ F y ∂ x − ∂ F x � � F x dx + F y dy = ∂ y dxdy ∂ D D y x Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 5 / 15

What is Green’s Theorem? Line integral ∆ i = ( x i + 1 − x i − 1 , y i + 1 − y i − 1 ) . . . . . . . . . . . . . . . . . . . ∆ 1 . ∆ 2 . ∆ 3 . . . . . . . . . . . . . ∆ 4 ∆ 5 Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 5 / 15

What is Green’s Theorem? Line integral n n � � F i • ∆ i = F x i ∆ x i + F y i ∆ y i 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∆ n F n F 1 ∆ 1 . . F 2 ∆ 2 . . F 3 ∆ 3 . . F 4 ∆ 4 . . . F 5 ∆ 5 . . . . . . . . . . . . . . . . . . . . . Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 5 / 15

What is Green’s Theorem? Line integral n n � � F i • ∆ i = F x i ∆ x i + F y i ∆ y i 1 1 This summation approximates: ◮ Rotation of a field ◮ Circulation of a fluid w.r.t. a boundary ◮ work done by a field on a particle Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 5 / 15

What is Green’s Theorem? Line integral When ∆ i → 0 (equiv. n → ∞ ) � 1 n � � F x i ∆ x i + F y i ∆ y i = F ≡ F x ( γ ( t )) γ ′ x ( t ) + F y ( γ ( t )) γ ′ y ( t ) dt γ 0 1 where the line is parametrised as γ : [ 0 , 1 ] ⇒ R 2 (i.e. 1-cube) In Isabelle/HOL, we defined it on top of Henstock-Kurzweil integral for: ◮ a field F : Euclidean Space ⇒ Euclidean Space. ◮ projected on a subset of the Basis of the space Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 5 / 15

What is Green’s Theorem? Double integral ∂ F y ∂ x − ∂ F x � � F x dx + F y dy = ∂ y dxdy ∂ D D y x Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 6 / 15

What is Green’s Theorem? Double integral ∆ i = ( x i + 1 − x i )( y i + 1 − y i ) . . . . . ∆ n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∆ 1 ∆ 2 ∆ 3 ∆ 4 ∆ 5 ∆ 6 Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 6 / 15

What is Green’s Theorem? Double integral The double integral can be approximated by the summation n � g ( x i , y i )∆ i i = 1 where g : R 2 ⇒ R is a “scalar” function. ◮ We used the Henstock-Kurzweil integral in Isabelle/HOL to model when n → ∞ ◮ In our case g = ∂ F y ∂ x − ∂ F x ∂ y , i.e. ◮ the rate of change of the line integral w.r.t. area of D ◮ models the vorticity of a fluid, or field rotation density, etc. Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 6 / 15

What is Green’s Theorem? Green’s Theorem: Applications ◮ in mathematical analysis, e.g. ◮ derive Cauchy’s integral theorem ◮ manipulating partial differential equations ◮ in analytical/mathematical physics, e.g. ◮ electromagnetism and electrodynamics: e.g. deriving Faraday’s law (point form) ◮ astronomy: e.g. deriving Kepler’s law for heavenly bodies ◮ Justification of efficient numerical methods for ◮ approximating integral on the boundary O ( n ) vs O ( n 2 ) ◮ fluid dynamics ◮ image processing Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 7 / 15

Traditional statement and proof of Green’s theorem Green’s Theorem: Elementary Regions D x is a type I region iff there are C 1 smooth functions g 1 and g 2 such that for two constants a and b : D x = { ( x , y ) | a ≤ x ≤ b ∧ g 2 ( x ) ≤ y ≤ g 1 ( x ) } g 1 ( x ) g 2 ( x ) x = a x = b Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 8 / 15

Traditional statement and proof of Green’s theorem Green’s Theorem: Elementary Regions D y is a type II region iff there are C 1 smooth functions g 1 and g 2 such that for two constants a and b : D y = { ( x , y ) | a ≤ y ≤ b ∧ g 2 ( y ) ≤ x ≤ g 1 ( y ) } y = b g 2 ( y ) g 1 ( y ) y = a Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 8 / 15

Traditional statement and proof of Green’s theorem Green’s Theorem: Elementary Regions ◮ D x is formalised as c : [ 0 , 1 ] 2 ⇒ R 2 (i.e. 2-cube), such that: c ▲ [ 0 , 1 ] 2 ▼ = { ( x , y ) | a ≤ x ≤ b ∧ g 2 ( x ) ≤ y ≤ g 1 ( x ) } ( 1 , 1 ) g 1 ( x ) c c ( 1 , 1 ) g 2 ( x ) x = a x = b ( 0 , 0 ) c ( 0 , 0 ) Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 8 / 15

Traditional statement and proof of Green’s theorem Green’s Theorem: Elementary Regions ◮ ∂ D x is the set of oriented paths (i.e. 1-chain) { ( − 1 , ( λ t . c ( 0 , t ))) , ( 1 , ( λ t . c ( 1 , t ))) , ( 1 , ( λ t . c ( t , 0 ))) , ( − 1 , ( λ t . c ( t , 1 ))) } g 1 ( x ) g 1 ( x ) g 2 ( x ) g 2 ( x ) x = a x = a x = b x = b Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 8 / 15

Traditional statement and proof of Green’s theorem Green’s Theorem: Elementary Regions ◮ ∂ D x is the set of oriented paths (i.e. 1-chain) { ( − 1 , ( λ t . c ( 0 , t ))) , ( 1 , ( λ t . c ( 1 , t ))) , ( 1 , ( λ t . c ( t , 0 ))) , ( − 1 , ( λ t . c ( t , 1 ))) } g 1 ( x ) g 1 ( x ) g 2 ( x ) g 2 ( x ) x = a x = a x = b x = b Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 8 / 15

Traditional statement and proof of Green’s theorem Green’s Theorem: Elementary Regions ∂ F y ∂ x − ∂ F x � � F x dx + F y dy = ∂ y dxdy ∂ D x D x Using ◮ line integral of F x on a vertical line is 0 ◮ Fubini’s theorem ◮ algebraic manipulation Where ◮ the line integral is lifted to 1-chains Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 8 / 15

Traditional statement and proof of Green’s theorem Green’s Theorem: Elementary Regions ∂ F y ∂ x − ∂ F x � � F x dx + F y dy = ∂ y dxdy ∂ D y D y Using ◮ line integral of F y on a vertical line is 0 ◮ Fubini’s theorem ◮ algebraic manipulation Where ◮ the line integral is lifted to 1-chains Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 8 / 15

Traditional statement and proof of Green’s theorem Green’s Theorem: Regions with piecewise smooth boundaries For a region D that can be divided in finitely many Type I 2-cubes (i.e. a type I 2-chain) Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 9 / 15

Traditional statement and proof of Green’s theorem Green’s Theorem: Regions with piecewise smooth boundaries For a region D that can be divided in finitely many Type I regions (i.e. a type I 2-chain) Mohammad Abdulaziz An Isabelle/HOL Formalisation of Green’s Theorem August 25, 2016 9 / 15