Multivariable Calculus Lecture #12 Notes

Transcription

1 Multivariable alculus Lecture #12 Notes In this lecture, we will develop a list of statements equivalent to a vector field being conservative and state and prove Green s heorem to help connect these facts. We ll also state and prove a Normal Form of Green s heorem that will be a two-dimensional preview of the ivergence heorem. We ll start by defining (both algebraically and geometrically) the divergence and curl of a vector field in R 3 as well as 2-dimensional versions of these that are relevant for Green s heorem (both versions). ivergence and curl of vector fields in R 2 and R 3 Suppose F ( xy, ) = Pxy (, ), Qxy (, ) defines a vector field in some region in R 2 where the component functions Pxy (, ) and Qxy (, ) are differentiable. We define: div( F ) = + (two-dimensional divergence of F) 2-curl( F ) = - (two-dimensional curl of F) hese are just formal algebraic definitions for these quantities. We will later redefine these in a coordinate-free, geometric manner and show that the above algebraic definitions are equivalent. If F ( xyz,, ) = Pxyz (,, ), Qxyz (,, ), Rxyz (,, ) defines a vector field in some region in R 3 where the component functions Pxyz (,, ), Qxyz (,, ) and Rxyz (,, ) are differentiable, we define: R div( F ) = + + (divergence of F) R R curl( F ) =,, (curl of F) he latter definition may be formally expressed in terms of a determinant as i j P Q R where we apply the derivatives appropriately to the component functions rather than calculate any actual products. Again, these are just formal algebraic definitions. We will later redefine these in a coordinate-free, geometric manner and show that the above algebraic definitions are equivalent. Note that if we display all possible partial derivatives of these component functions in the 3 3 matrix, then the divergence is just the trace of this matrix (sum of the main diagonal entries) and the R R R curl is constructed (in perhaps a mysterious way) from the remaining six entries. It must be emphasized that the divergence of a vector field is a scalar-valued function, and the curl of a vector field is also a vector field. Equivalent statements to a vector field being conservative Suppose that F = PQ, defines a vector field in some simply connected region in R 2 where the component functions P and Q are differentiable; or that F = PQR,, defines a vector field in some region in R 3 where 1

2 the component functions P, Q and R are differentiable. [A region is called simply connected if any closed path (loop) with the region can be continuously contracted down to a single point while remaining in the region.] hen the following statements are equivalent: (1) F is conservative. (2) he wor integral F dr is independent of the path between two fixed points A and B in the region. (3) he circulation F d r = 0 around any closed path (loop) in the region. (4) F = V for some differentiable function V. [ Vxy (, ) in the R 2 case, and Vxyz (,, ) in the R 3 case.] (5) 2-curl( F ) = - = 0 or, equivalently, = in the R 2 case, R R curl( F) =,, = 0 or, equivalently, = x, R = x, and R = in the R 3 case. We previously referred to this (these) condition(s) as the test for exactness. (1) and (2) are equivalent by the definition of a conservative vector field. It s easy to see that (2) implies (3) by inserting any two points along the closed curve and noting that the wor will be the same following the two possible routes between these points together with the fact that following one in reverse will simply reverse the sign for the wor. his argument is reversible, so we also have that (3) implies (2). Gradient implies conservative: We have already shown that (4) implies (1) by the Fundamental heorem of Line Integrals, and that (4) implies (5) by lairaut s heorem. onservative implies gradient: o show that (1) implies (4), suppose F is conservative. We ll do this in the R 3 case. Pic any fixed point ( the ground ) ( x0, y0, z 0) in the given region and for any other point ( xyz,, ) in the region define V ( x, y, z ) = F d r = Pdx + Qdy + Rdz where is any curve from ( x 0, y0, z 0) to ( xyz,, ). his is well-defined because the vector field F is presumed to be conservative. We ll show that = P. he calculation for the other components is similar. Suppose we vary x only by an amount x starting at ( xyz,, ) and ending at ( x+ xyz,, ). Over this segment we ll have dy = 0 and dz = 0, so the change V will be given by V = Vx ( + xyz,, ) Vxyz (,, ) = F d r = Pdx where is the aforementioned short segment. he integral is given approximately by Pxyz (,, ) x where x is between x and x+ x. So we have V= Vx ( + xyz,, ) Vxyz (,, ) Pxyz (,, ) x. ivision by x then gives V Vx ( + xyz,, ) Vxyz (,, ) = Pxyz (,, ), and if we then pass to the limit as x 0, x will be squeezed x x V Vx ( + xyz,, ) Vxyz (,, ) toward x and we ll have = lim = Pxyz (,, ) x 0 x. Similarly, = Q and = R. All that s left to prove is that (5) implies any of the other statements. We ll show that (5) implies (3), but to do so will require another theorem or two. In the R 2 case we ll need Green s heorem, and in the R 3 case we ll need Stoes heorem both of which are, in fact, just different versions of the Fundamental heorem of alculus. 2

3 Green s heorem: Suppose F ( xy, ) = Pxy (, ), Qxy (, ) defines a vector field in some bounded region in R 2 where the component functions Pxy (, ) and Qxy (, ) are differentiable. Let be the boundary of this region oriented in the counterclocwise sense (this can be understood generally to mean that as you transfer the boundary the region will always be to the left). We denote this by Bnd( ) = =. hen: circulation of = d = Pdx + Qdy = da around = = = F r F In other words, the circulation around the boundary is the same as the integral of the 2-curl over the interior. All versions of the Fundamental heorem of alculus share this same theme of trading in a boundary for some ind of derivative and integrating over the interior. orollary: Suppose F ( xy, ) = Pxy (, ), Qxy (, ) defines a vector field in some bounded, simply connected region in R 2 where the component functions Pxy (, ) and Qxy (, ) are differentiable and let Bnd( ) =. If = through the region, then 0 F d r =. hat is, (5) implies (3). Proof: If = y, then = 0 throughout. herefore F dr = da = 0dA = 0 =. We ll delay the proof that (5) implies (3) in the R 3 case until we state and prove Stoes heorem. Geometric, coordinate-free definition of 2-curl: he curl of a vector field can best be understood as a 2 circulation density. If we choose any point ( xy, ) R, let be any (small) region that contains this point and let = be its boundary (oriented in the counterclocwise sense). We define the circulation density of F at the point ( xy, ) by calculating the circulation of F about = as a fraction of the Area( ) = A of the small region, and then tae the limit as this small region shrins down to the single point ( xy, ) - assuming, for the moment, that this limit exists independent of any choices during the construction. hat is: d [2-curl( )]( xy, ) lim F r = F = diam( ) 0 A Proof of Green s heorem: Partition the region into small cells and let = be the boundary of the -th cell. hoose a sample point ( x, y) for each cell. hen from the geometric limit definition above we F dr = can say that for all, [2-curl( F)]( x, y), so {[2-curl( )](, )} A F dr F x y A. = F dr F x y A. = Summing over we have that ( ) {[2-curl( )](, )} Observe that in the left-hand sum the contribution from any adjacent cells will cancel pairwise since those portions of the boundaries will be in opposite directions. herefore the only contributions will be from the cells with boundaries on the overall boundary of the region. hat is, F dr { [2-curl( F)]( x, y) } A. = 3

4 Finally, by refining the partition and passing to the limit as the mesh of the partition tends to zero, the approximation will approach an equality, so we ll have: F dr = lim { [2-curl( F)]( x, y )} A = {[2-curl( F)]( x, y) } da = 0 Basically, the proof of Green s heorem is really just a corollary of the geometric definition of the curl. here is one missing piece that we still need to resolve namely that the geometric definition of the 2-curl implies the algebraic definition, i.e. 2-curl( F ) = -. o do this, recall that the wor integral around any small cell is F dr = F ds = where F = F and is the unit tangent vector for the boundary = curve. o derive the artesian expression for the 2-curl, we choose a small rectangular cell with side lengths x and y and, for convenience, locate the point ( xy, ) at the (lower left) corner of this cell. he integral can then be approximated by adding up the contributions from the four sides of this cell. We can assemble the necessary information in a convenient table: Side F = F s Bottom i Pxy (, ) x Right j Qx ( + xy, ) y op i Pxy (, + y) x Left j Qxy (, ) y For each side we chose the most convenient point on that side for our approximate values. If we sum these four contributions and divide by the area of the cell, we get: F ds [ (, ) (, )] [ (, ) (, )] = F s Qx+xy Qxy y Pxy+y Pxy x = A A x y [ Qx ( +xy, ) Qxy (, )] [ Pxy (, +y) Pxy (, )] = x y Finally, if we pass to the limit as both x and y approach zero, we ll have: F ds [ (, ) (, )] [ (, ) (, )] [2-curl]( ) lim = Qx+xy- Qxy Pxy+y-Pxy F = = lim - lim = - 0 A x 0 y 0 x y urious orollary of Green s heorem: Suppose is any region in the xy-plane and that = is its boundary. hen Area( ) = xdy. = As a practical matter, this means that we can either parameterize the boundary curve and calculate the given integral OR choose closely-spaced waypoints ( x, y ) all along the boundary and approximate the integral by calculating the sum x y. his is relatively easy to carry out using a GPS device. Proof of orollary: xdy = 0 dx + xdy = (1 0) da = da = Area( ) = = 4

5 Normal Form of Green s heorem he standard form of Green s heorem is derived by considering the wor integral around the counterclocwise boundary of a region, i.e. F ds = ds F. We can alternatively rotate the unit tangent vector = = clocwise 90 to produce an outward unit normal vector N at every point of this curve (or any curve). For a small segment of the boundary with length s, we can measure the flux (or flow) of the vector field across the curve as FN s where F N = F N is the outward normal component of the vector field at any given point. If we sum these up over the entire curve and pass to the limit as these pieces become arbitrarily small, we can define the flux of F across the curve as F ds = ds N F N. In artesian coordinates, if F ( xy, ) = Pxy (, ), Qxy (, ) and if we parameterize the curve as r () t = xt (), yt (), dx dy dx dy then v () t =, and formally ds dt dt, dt dx, dy d dt dt = v v = v = dt dt = = r v dy dx If we rotate this clocwise 90, we get N ds =, dt = dy, dx, so: dt dt F ds = F N ds = P, Q dy, dx = Q( x, y) dx + P( x, y) dy N Normal Form of Green s heorem: If F ( xy, ) = Pxy (, ), Qxy (, ) defines a vector field in some bounded region in R 2 where the component functions Pxy (, ) and Qxy (, ) are differentiable and where = is the boundary of this region oriented in the counterclocwise sense, then: net flux of F outward = FNds = Qdx + Pdy = + da = div( ) da across = = = F where div( F ) is the two-dimension divergence of this vector field. his provides some explanation for the interpretation of the divergence of a vector field as a source density. Essentially, the total amount of stuff flowing outward across the boundary of a closed region should measure the total amount of the source of that stuff emanating from within the region. he three-dimensional version of this will be the ivergence heorem. In the next lecture we ll loo at integration on surfaces. Notes by Robert Winters and Renée hipman 5