From Math 2220 Class 40 Integration and Stokes for Differential - PowerPoint PPT Presentation

From Math 2220 Class 40 V1df Algebra of Differential Forms From Math 2220 Class 40 Integration and Stokes for Differential Forms Dr. Allen Back Stokes and Gauss Traditionally Dec. 3, 2014

Algebra of Differential Forms Differential forms are sums of wedge products of the “basis From Math 2220 Class 40 1-forms” dx , dy , and dz . They are kinds of tensors generalizing V1df ordinary scalar functions and vector fields. They have a Algebra of “skew-symmetry” property so that for 1 − forms (e.g. dx or dy ) Differential Forms Integration ω ∧ η = − η ∧ ω. and Stokes for Differential Forms Stokes and This implies dx ∧ dx = 0 for example. Gauss Traditionally Marsden and Tromba drop the ∧ symbol, writing dz dx for dz ∧ dx , etc.

Algebra of Differential Forms In R 3 all differential forms have an interpretation as either an From Math 2220 Class 40 ordinary function or a vector field. V1df 0-forms are scalar functions - f ( x , y , z ) on R 3 . Algebra of 1-forms are linear combinations P dx + Q dy + R dz Differential Forms where P , Q , and R are functions of three Integration variables ( x , y , z ) . Think of this as associated and Stokes for Differential with the vector field ( P , Q , R ) . Forms 2-forms are linear combinations Stokes and Gauss Traditionally P dy ∧ dz + Q dz ∧ dx + R dx ∧ dy . Think of this as associated with the vector field ( P , Q , R ) . 3-forms are of the form f ( x , y , z ) dx ∧ dy ∧ dz . Think of this as associated with the scalar function f ( x , y , z ) .

Algebra of Differential Forms From Math 2220 Class 40 There is an operation exterior differentiation d which takes any V1df k form to a k + 1 form. Algebra of Differential For ordinary functions f ( x , y , z ) , it looks like the expression for Forms differentials: Integration and Stokes for Differential Forms df = f x dx + f y dy + f z dz . Stokes and Gauss Traditionally

Algebra of Differential Forms From Math 2220 Class 40 d 2 = 0; i.e. for any ( C 2 ) k -form ω V1df Algebra of Differential d ( d ω ) = 0 Forms Integration and Stokes for So e.g. d ( dx ) = 0 . Differential Forms In the end this is because of the independence of order for Stokes and mixed partials. Gauss Traditionally

Algebra of Differential Forms From Math Exterior derivative obeys the following anti-commutativity rule 2220 Class 40 for wedge products: V1df Algebra of Differential d ( η ∧ ω ) = ( d η ) ∧ ω + ( − 1) k η ∧ ( d ω ) . Forms Integration and Stokes for Differential if η is a k -form and ω is an l -form. Forms Stokes and Gauss ω ∧ η = ( − 1) kl η ∧ ω as well. Traditionally

Algebra of Differential Forms For differential forms, the operations of grad, div, and curl are From Math 2220 Class 40 all special cases of exterior differentiation d ! V1df grad is d from 0-forms (functions) to 1-forms as long Algebra of Differential as we identify the 1-form P dx + Q dy + R dz Forms with the vector field ( P , Q , R ) . Integration and Stokes for curl is d from 1-forms to 2-forms as long as we Differential Forms identify the 1 and 2-forms with their associated Stokes and vector fields. Gauss Traditionally div is d from 2-forms (vector fields in disguise) to 3-forms (functions in disguise.)

Algebra of Differential Forms From Math Let D = [0 , 1] × [0 , 1] ⊂ R 2 be a square, Φ : D → R 3 a 2220 Class 40 parameterization of a piece of surface S , and ω a 1-form on R 3 . V1df Algebra of Differential There is a pullback operation Φ ∗ taking a 2-form (or more Forms Integration generally any k -form) on the image of Φ to a 2-form on R 2 . and Stokes for Differential Forms Φ( u , v ) = ( u 3 + v 3 , u − v , u + v ) , Stokes and Gauss Traditionally Φ ∗ ( xy dx ∧ dy ) = ( u 3 + v 3 )( u − v ) (3 u 2 du + 3 v 2 dv ) ∧ ( du − dv ) .

Integration and Stokes for Differential Forms From Math 2220 Class 40 Integration of a differential k -form over a rectangle in R k is the V1df Algebra of same as an ordinary k -fold multiple integral. Differential Forms We use chains (essentially formal sums of parameterizations, Integration with coefficients like ± 1 to express orientation) to reduce the and Stokes for Differential integration of any differential k -form (over e.g. a path, surface, Forms or region in R 3 ) to an ordinary multiple integral. Stokes and Gauss Traditionally

Integration and Stokes for Differential Forms Let D = [0 , 1] × [0 , 1] be a square, Φ : D → R 3 a From Math 2220 Class 40 parameterization of a piece of surface S , and ω a 1-form on R 3 . V1df Algebra of Differential Assume Φ( ∂ D ) = ∂ (Φ( D )) ∂ S . Then Forms � � � Integration ω = d ω and Stokes for Differential Φ( ∂ D ) Φ( D ) Forms Stokes in R 3 follows from Stokes and Gauss Traditionally � � � Φ ∗ ω = d Φ ∗ ω ∂ D D on the square in R 2 . (A simple version of Green.)

Integration and Stokes for Differential Forms From Math 2220 Class 40 V1df Here Φ ∗ refers to the pullback operation; e.g. for Algebra of Differential Forms Φ( u , v ) = ( u 3 + v 3 , u − v , u + v ) , Integration and Stokes for Φ ∗ ( xy dx ∧ dy ) = ( u 3 + v 3 )( u − v ) (3 u 2 du + 3 v 2 dv ) ∧ ( du − dv ) . Differential Forms Stokes and Gauss Traditionally

Integration and Stokes for Differential Forms Integration of differential forms is basically defined in terms of From Math 2220 Class 40 pullback. V1df If c : [ a , b ] → R 3 with c ( t ) = ( x ( t ) , y ( t ) , z ( t )), then Algebra of c ∗ dx = x ′ ( t ) dt , etc. and so the integral of the 1-form Differential Forms η = P dx + Q dy + R dz over C corresponding to c is Integration essentially defined by and Stokes for Differential � � Forms η = P dx + Q dy + R dz Stokes and C C Gauss Traditionally � c ∗ ( P dx + Q dy + R dz ) = C � b ( Px ′ + Qy ′ + Rz ′ ) dt = a in line with one of our older classical notations/computation styles for line integrals.

Integration and Stokes for Differential Forms From Math 2220 Class 40 V1df Now let’s look at the differential form version of the surface Algebra of integral Differential Forms �� ( x , y , z ) · ˆ n dS Integration and Stokes for S Differential Forms over part of the paraboloid z = 1 − x 2 − y 2 with z ≥ 0 . Stokes and Gauss Traditionally

Integration and Stokes for Differential Forms From Math 2220 Class 40 The associated 2-form to the vector field ( x , y , z ) is V1df µ = x dy ∧ dz − x dz ∧ dx + z dz ∧ dy . Algebra of Differential Forms Integration and Stokes for For the parameterization Differential Forms c ( r , θ ) = ( r cos θ, r sin θ, 1 − r 2 ) , Stokes and Gauss Traditionally we have dy = cos θ dr − r sin θ d θ , dz = − 2 r dr etc., so

Integration and Stokes for Differential Forms From Math 2220 Class 40 V1df �� µ = Algebra of S Differential �� Forms c ∗ ( x dy ∧ dz − x dz ∧ dx + z dx ∧ dy ) = Integration D and Stokes for � 1 � 2 π Differential Forms = r cos θ (cos θ dr − r sin θ d θ ) ∧ ( − 2 r dr ) + . . . Stokes and 0 0 Gauss Traditionally which (thinking about ˆ n in terms of cross products) can be seen to agree with our usual!

Stokes and Gauss Traditionally From Math 2220 Class 40 V1df Integration of a conservative vector field cartoon. Algebra of Differential Forms Integration and Stokes for Differential Forms Stokes and Gauss Traditionally

Stokes and Gauss Traditionally From Math 2220 Class 40 V1df Green’s Theorem cartoon. Algebra of Differential Forms Integration and Stokes for Differential Forms Stokes and Gauss Traditionally

Stokes and Gauss Traditionally From Math 2220 Class 40 V1df Stokes’ Theorem cartoon. Algebra of Differential Forms Integration and Stokes for Differential Forms Stokes and Gauss Traditionally

Stokes and Gauss Traditionally From Math Both sides of Stokes involve integrals whose signs depend on 2220 Class 40 the orientation, so to have a chance at being true, there needs V1df to be some compatibility between the choices. Algebra of Differential Forms The rule is that, from the “positive” side of the surface, (i.e. Integration the side chosen by the orientation), the positive direction of the and Stokes for Differential curve has the inside of the surface to the left. Forms Stokes and As with all orientations, this can be expressed in terms of the Gauss Traditionally sign of some determinant. (Or in many cases in terms of the sign of some combination of dot and cross products.)

Stokes and Gauss Traditionally From Math 2220 Class 40 Problem: Let S be the portion of the unit sphere V1df x 2 + y 2 + z 2 = 1 with z ≥ 0 . Orient the hemisphere with an Algebra of F ( x , y , z ) = ( y , − x , e z 2 ) . Calculate upward unit normal. Let � Differential Forms the value of the surface integral Integration and Stokes for Differential �� ∇ × � Forms F · ˆ n dS . Stokes and S Gauss Traditionally

Stokes and Gauss Traditionally From Math 2220 Class 40 V1df Gauss’ Theorem field cartoon. Algebra of Differential Forms Integration and Stokes for Differential Forms Stokes and Gauss Traditionally