18 02 Multivariable Calculus Spring 2009 Lecture 27 Simply Connected Regions April 14 Reading Material From Lecture Notes V5 and V6 Last time Green s Theorem why it works Examples Today Simply Connected Regions 2 Generalized or Extended Green s Theorem Consider a region R with n holes Call C0 the outside boundary and C1 Cn the inside boundaries of the holes Orient the boundaries so that the region R is always on the left while wolking on these curves Also assume that these curves are simple and closed Assume that a vector filed F is continuously differentiable on R Then the Green s theorem says that ZZ n I X N M F d r dA x y R i 0 Ci Proof Let s consider for simplicity a region with two holes We cut through them and we split the region R into two new regions R1 and R2 that are now without holes Let s call their boundaries B1 and B2 respectively 1 For each of the two regions we apply the usual Green s theorem I F d r B1 I B2 F d r ZZ R1 M N x y ZZ dA R2 M N x y ZZ dA R M N x y dA On the other hand in the left hand side since the line integral on the segment introduced in the cutting appears twice but in opposite directions we have that the only contribution remaining is relative only to the outside boundary C0 and the boundaries of the two holes C1 and C2 this concludes the proof 3 Simply connected regions Consider the two vector fields G F x y i 2 j x2 y 2 x y2 x y i 2 j 2 x x 3 1 3 2 Question Are these two vectors fields conservative in their domains that we already discussed It s domain is the whole space minus We first consider the vector field G 0 0 We know that x2 y 2 My 2 Nx 3 3 x y 2 2 is conservative since we already but in this case having zero curl it is not enough to claim that G easily computed that I d r 2 G 3 4 C cannot be where C is any circle centered at 0 0 oriented counterclockwise Since C is closed G conservative We can remark though using the generalized Green s theorem that any other simple closed curve C containing 0 0 oriented counterclockwise is such that I d r 2 G C 2 To see this take a little circle cr of radius r centered at 0 0 and contained inside C Orient cr clockwise and let R be the region below Then I d r G C I d r G ZZ cr R M N x y dA but the Right Hand Side is zero by 3 3 and by 3 4 we obtain I I d r 2 G d r G C cr We consider now the vector filed F Its domain is the whole plane minus the y axis x 0 We also have in this domain that 1 Nx 3 5 x2 So in each half plane we can actually use a generalization of the Gradient test see Theorem 1 below to show that each closed simple curve in the domain has zero line integral One can also find a potential function y f x y C x and F is that the first one is connected The difference between the domains of the vector field G but not simply connected and the second one is made of two disconnected pieces both of them simply connected accordingly to the following definition My Definition 1 A two dimensional region D of the plane consisting of one single piece is called simply connected if it has this property whenever a simple closed curve C lies entirely in D then its interior part also lies entirely in D 3 Examples of non simply connected regions Examples of simply connected regions We have the following theorem Theorem 1 Generalized Gradient Test If F is continuously differentiable on a simply connected region then F is conservative if and only if Nx My 0 Moreover there exists a potential function f for F Intuition Of course in this case one can repeat the proof that was given last time by cutting the region into small rectangles The point is that by definition each rectangular curve that sits inside the simply conncetd region has all its interior inside it too On the other hand consider a vector field that is singular at a point its domain is then not simply connnected We can take a curve around that point where the vector field is well defined It is true that zero curl Nx My 0 inside the curve implies nothing going around hence no work along the curve but we don t know what happens at the missing point and that could account for work done on the curve In a similar language but using flux what we just said can be rephrased in the following way consider a vector filed that is singular at a point and has zero divergence everywhere else Mx Ny 0 Again let s pick a curve around the singular point Of course zero divergence inside the region enclosed by the curve means that there are no sources and no sinks but we don t know what happens at the singular point That point may have a source or sink and that could account for flux trough the curve Exercise 1 Consider the vector filed F 2x 1 i j y x2 2 2 y x2 2 2 Find its domain D and decide if it is conservative or not 4 Solution We observe that D is the whole plane minus the parabola y x2 2 We observe that D D1 D2 and both D1 and D2 are simply connected regions We then look at the curl of F and we see that 4x N x My y x2 2 3 hence the generalized gradient test shows that F is conservative on each region D1 and D2 and hence on D One can also easily show that f x y 1 y x2 2 is a potential function 4 What we did in 2D and what we need to do in 3D 2D 3D Double integrals Triple integrals Polar coordinates Cylindrical spherical coordinates 2D Applications 3D Applications Line integrals in 3D Line integrals Surface integrals Green s Theorem for flux Gauss divergence Theorem Green s Theorem for work Stoke s Theorem 5 Introduction to triple integrals The 2D case We recall that given a function f x y and a region R 5 we can write ZZ Z b Z g x f x y dy dx f x y dA R h x a To determine the limits of the other variable we look at the shadow of R on the axis relative to the other variable The 3D case Here we usually have a function f x y z and a 3D region D and we want to define ZZZ f …
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