MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use visit http ocw mit edu terms V14 Some Topological Questions We consider once again the criterion for a gradient field We know that and inquire about the converse It is natural to try to prove that by using Stokes theorem if curl F 0 then for any closed curve C in space The difficulty is that we are given C but not S So we have to ask Question Let D be a region of space in which F is continuously differentiable Given a closed curve lying in D is it the boundary of some two sided surface S lying inside D We explain the two sided condition Some surfaces are only one sided if you start painting them you can use only one color if you don t allow abrupt color changes An example is S below formed giving three half twists to a long strip of paper before joining the ends together Surface S Boundary C trefoil knot S has only one side This means that it cannot be oriented there is no continuous choice for the normal vector n over this surface If you start with a given n and make it vary continuously when you return to the same spot after having gone all the way around you will end up with n the oppositely pointing vector For such surfaces it makes no sense to speak of the flux through S because there is no consistent way of deciding on the positive direction for flow through the surface Stokes theorem does not apply to such surfaces To see what practical difficulties this causes even when the domain is all of 3 space consider the curve C in the above picture It s called the trefoil knot We know it is the boundary of the one sided surface S but this is no good for equation 3 which requires that we find a two sided surface which has C for boundary There are such surfaces try to find one It should be smooth and not cross itself If successful consider yourself a brown belt topologist The preceding gives some ideas about the difficulties involved in finding a two sided surface whose boundary is a closed curve C when the curve is knotted i e cannot be continuously deformed into a circle without crossing itself a t some point during the deformation It is by no means clear that such a two sided surface exists in general There are two ways out of the dilemma V VECTOR INTEGRAL CALCLUS 2 1 If we allow the surface to cross itself and allow it to be not smooth along some lines we can easily find such a two sided surface whose boundary is a given closed curve C The procedure is simple Pick some fixed point Q not on the curve C and join it to a point P on the curve see the figure Then as P moves around C the line segment Q P traces out a surface whose boundary is C It will not be smooth at Q and it will cross itself along a certain number of lines but it s easy to see that this is a two sided surface Q P The point now is that Stokes theorem can still be applied to such a surface just use subdivision Divide up the surface into skinny triangles each having one vertex at Q and include among the edges of these triangles the lines where the surface crosses itself Apply Stokes theorem to each triangle and add up the resulting equations 2 Though the above is good enough for our purposes it s an amazing fact that for any C there is always a smooth two sided surface which doesn t cross itself and whose boundary is C This was first proved around 1930 by van Kampen I I The above at least answers our question affirmatively when D is all of 3 space Suppose however that it isn t If for instance D is the exterior of the cylinder x 2 y2 1 then it is intuitively clear that a circle C around the outside of this cylinder isn t the boundary of any finite surface lying entirely inside D A class of domains for which it is true however are the simply connected ones Definition A domain D in 3 space is simply connected if each closed curve in it can be shrunk to a point without ever getting outside of D during the shrinking For example 3 space itself is simply connected as is the interior or the exterior of a sphere However the interior of a torus a bagel for instance is not simply connected since any circle in it going around the hole cannot be shrunk to a point while staying inside the torus If D is simply connected then any closed curve C is the boundary of a two sided surface which may cross itself lying entirely inside D We can t prove this here but it gives us the tool we need to establish the converse to the criterion for gradient fields in 3 space Theorem Let D be a simply connected region in 3 space and suppose that the vector field F is continuously differentiable in D Then in D 5 curlF O F Vf Proof According to the two fundamental theorems of calculus for line integrals section V11 3 it is enough to prove that F dr 0 for every closed curve C lying in D Since D is simply connected given such a curve C we can find a two sided surface S lying entirely in D and having C as its boundary Applying Stokes theorem h d L c u r l d 0 which shows that F is conservative and hence that F is a gradient field Summarizing we can say that if D is simply connected the following statements are equivalent if one is true so are the other two rQ F Vf curlF O H Jp F dr i s path independent I V14 SOME TOPOLOGICAL QUESTIONS 3 Concluding remarks about Stokes theorem Just as problems of sources and sinks lead one to consider Green s theorem in the plane for regions which are not simply connected it is important to consider such domains in connection with Stokes theorem For example if we put a closed loop of wire in space the exterior of this loop the region consisting of 3 space with the wire removed is not simply connected If the wire carries current the resulting electromagnetic force field F will satisfy curl F 0 but F will not be conservative In particular the value of f F dr around a closed path which links with the loop will not in general be zero which explains why you can get power from a wire carrying current even though the curl of its electromagnetic field is zero As an example consider the vector field in 3 space The domain of definition is xyz space with the z axis removed …
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