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Line IntegrationLast time: a 3D vector field is conservative if: ________________________a 2D vector field is conservative if: ________________________Line IntegrationWe now introduce a new type of integration that will come in handy when we integrate vector fields. For now, we define:Definition: If rt=<xt,yt,zt>, for a≤t≤b is a curve C, and f(x,y,z) a function defined in a neighborhood of the curve C, then we define the (line) integral of the function f over the curve C as:Cf(x,y)ds=abf(xt,yt)x't2+y't2dtNote: If ft=1 then C1ds=abx't2+y't2dt is the: __________________Example: Find Cxy2ds, where C is the straight line from (-1,-1) to (1,3).To compute line integrals we need to be familiar with “paths”. The following paths occur frequently:line segment from P to Qcircle around 0, radius Rgraph of y=f(x)graph of x=g(y)Example: Find C2xds, where C is a portion of a standard parabola from (0,0) to (1,1) followed by a line segment from (1,1) to (2,3). Also, find Cds, where C is the line segment from (-1,2) to (2,-1)Other line integralsCf(x,y)ds=abf(xt,yt)x't2+y't2dt Cf(x,y)dx=abf(xt,yt)x'tdt (with respect to x)Cf(x,y)dy=abf(xt,yt)y'tdt (with respect to y) Cf(x,y)dx+gx,ydy=abf(xt,yt)x't+gxt,yty'(t)dtabf(x)dx Rf(x,y)dA Example: If the curve C is a straight line from (0,1) to (2,5), find the following line integrals:Cxy2ds Cxy2dx Cxy2dy Cxy2dx+2xdy Definition: The line integral of vector field F=<M,N,P> over a curve C given by rt=<xt,yt,z(t)> is:CF?dr=C<M,N,P>?<dx,dy,dz>=CMdx+Ndy+PdzThat integral gives the work necessary to move a particle through the field along the path CNote: CMdx+Ndy+Pdz=CM(xt,yt,zt x'(t)+N(xt,yt,zt y'(t)+P(xt,yt,zt z'(t)dtExample: Find the work necessary to move a particle in a straight line from (0,1) to (3,4) through the field Fx,y=<x2+y2,2xy>Find C<y2,x> ?dr, where (i) C1 is a line from (-5,-3) to (0,2) and for (ii) C2 is given by x=4-y2 from (-5,-3) to (0,2)Find the work necessary to move a particle along the line from (2,0,0) to (3,4,5) through F=<y,z,x>Next we will tie everything together by looking at the work done through conservative vector fieldsFundamental Theorem of Line IntegrationIf F is a conservative vector field with potential function f, and C is a curve from P to Q, then:CF?dr=fQ-f(P)Thus, if a vector field is conservative, we have two ways to find the work integral CF?dr: (i) you can use the definition of the line integral (as long as the path is explicitly given), or (ii) you could find the potential function and then compute the difference f(end) – f(start). Sometimes one is easier, sometimes the other:Name: ________________________Quiz 11Mat0063h the following vector fields: For the vector field Fx,y,z=<x2,2xy,z2-xy>, finddiv(F)curl( F)Which of the following vector fields are conservative? Find their potential function(s) if they are.F=<x3-2x2y3,y2-2x3y2>G=<xz,z+y,yz> ................
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