From Math 2220 Class 41 Uniqueness of Laplace Eqn Solutions - PowerPoint PPT Presentation

From Math 2220 Class 41 V1 Conservative Fields More Carefully From Math 2220 Class 41 Uniqueness of Laplace Eqn Solutions Maxwell’s Dr. Allen Back Equations to Light Integral Theorem Problems Dec. 5, 2014 Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 In order that a vector field ( P ( x , y ) , Q ( x , y )) be ∇ f = ( f x , f y ) V1 for a C 2 function f , the equality of mixed partials shows that Conservative Fields More we have a necessary condition Carefully Uniqueness of P y = Q x . Laplace Eqn Solutions Maxwell’s Equations to Light (This can also be phrased as the scalar curl Q x − P y = 0 . ) Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 V1 � − y dx + x dy The “ d θ ” example � = 0 for a circle enclosing x 2 + y 2 Conservative C Fields More the origin shows that this necessary condition is not always Carefully sufficient to guarantee the existence of such an f . Uniqueness of Laplace Eqn Here the vector field has domain R 2 − { (0 , 0) } , a region with a Solutions (tiny) hole in it. Maxwell’s Equations to Light Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 V1 Similarly in R 3 , the identity curl ( ∇ f ) = 0 for a C 2 vector field Conservative makes curl ( � F ) = 0 a necessary condition for being able to find Fields More Carefully a C 2 f with Uniqueness of ∇ f = � Laplace Eqn F . Solutions Maxwell’s Equations to Light Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 The vector field V1 Conservative � � 1 Fields More � F ( x , y , z ) = ( − y , x , 0) Carefully x 2 + y 2 Uniqueness of Laplace Eqn with domain R 3 − { (0 , 0 , z ) } (a region with a (thin) hole in it) Solutions shows that again curl ( � Maxwell’s F ) = 0 is not sufficient for the existence Equations to Light for such an f . Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math There is a topological condition on the domain D that does 2220 Class 41 guarantee these necessary conditions are sufficient; namely that V1 the domain of the vector field be simply-connected. Conservative Fields More Intuitively, D being simply connected means any closed curve C Carefully can be continuously shrunk down to a point (entirely within Uniqueness of Laplace Eqn D . ) (The region D is also assumed as part of simple Solutions connectivity to have just one piece; i.e. to be connected.) Maxwell’s Equations to Regions like all of R n , a ball, a rectangle, or any “convex” set Light are simply connected. Integral But a ring, R 2 − { (0 , 0) } , and R 3 − { (0 , 0 , z ) } are not. Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 V1 Conservative For a simply connected region, D , the necessary conditions are Fields More sufficient for the existence of such an f with ∇ f = � Carefully F . Uniqueness of Though doing this rigorously requires more topology than we Laplace Eqn Solutions have, the basic idea is to use Stokes theorem. Maxwell’s Equations to Light Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 V1 The idea is to pick a point p 0 ∈ D and define f ( p ) as follows: Conservative For any p ∈ D , pick a curve C from p 0 to p . Define Fields More Carefully Uniqueness of � � Laplace Eqn f ( p ) = F · d � s . Solutions C Maxwell’s Equations to Light Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 V1 Conservative Fields More Carefully Uniqueness of Laplace Eqn Solutions Maxwell’s Equations to Light Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 This apparently strange definition works better than you would V1 at first think because if we choose a different path C ′ from p 0 Conservative to p , for a simply connected region, we have the “path Fields More Carefully independence” property Uniqueness of Laplace Eqn Solutions � � � � F · d � s = F · d � s . Maxwell’s Equations to C ′ C Light Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 V1 Conservative Fields More Carefully Uniqueness of Laplace Eqn Solutions Maxwell’s Equations to Light Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 The idea is, for a simply connected region, we can, in a sense V1 good enough for Stokes, find a surface S filling in the inside of the closed loop C followed by C ′ in the reverse direction. Conservative Fields More Applying Stokes to this surface in the presence of curl ( � Carefully F ) = 0 Uniqueness of gives Laplace Eqn Solutions �� � � Maxwell’s curl ( � � � 0 = F ) · ˆ n dS = F · d � s − F · d � s . Equations to Light S C C ′ Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 V1 Conservative Fields More Carefully Uniqueness of Laplace Eqn Solutions Maxwell’s Equations to Light Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 V1 Conservative Fields More In the presence of path independence, we can argue (by using Carefully well chosen paths) why ∇ f = � F with this definition of f : Uniqueness of Laplace Eqn Solutions Maxwell’s Equations to Light Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 V1 The computational scheme we looked at before Thanksgiving for finding potentials is an instance of this procedure, even Conservative Fields More though it didn’t look that way. Carefully For example, with p 0 = (0 , 0 , 0) , picking the path C to consist Uniqueness of Laplace Eqn of 3 segments parallel to the coordinate axis,makes the line Solutions integral computation a sequence of 3 one-variable Maxwell’s Equations to anti-differentiations. Light Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 V1 Conservative Fields More Carefully Uniqueness of Laplace Eqn Solutions Maxwell’s Equations to Light Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 V1 There is a similar story with the converse to div ( curl ( � A )) = 0 Conservative for C 2 vector fields. Fields More Carefully A � A satisfying curl ( � A ) = � F is called a vector potential for � F . Uniqueness of They are widely used in studying magnetic fields � Laplace Eqn B , which by Solutions one of Maxwell’s equations satisfy div ( � B ) = 0 . Maxwell’s Equations to Light Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of

Conservative Fields More Carefully From Math 2220 Class 41 The point charge field V1 Conservative � r Fields More � F ( x , y , z ) = Carefully r � 3 � � Uniqueness of Laplace Eqn is an example of a vector field on R 3 minus the origin whose Solutions Maxwell’s divergence vanishes, yet (because of Gauss’ theorem applied to Equations to a ball around the origin) cannot be curl ( � Light A ) . Integral Theorem Problems Conservative Vector Fields Systematic Method of Finding a Potential Stokes and Gauss Surface of