Topic 5.5: Green s Theorem

Transcription

1 Math 275 Notes Topic 5.5: Green s Theorem Textbook Section: 16.4 From the Toolbox (what you need from previous classes): omputing partial derivatives. Setting up and computing double integrals (this includes finding limits of integration). Setting up and evaluate line integrals (this includes finding a parameterization). Learning Objectives (New Skills) & Important oncepts Learning Objectives (New Skills): Apply Green s Theorem to evaluate line integrals using a double integral. Recognize when Green s Theorem can be used. Important oncepts: Suppose is a closed curve in the plane, oriented counterclockwise, and is the planar region inside (we say: is the boundary of, or: is bounded by.). Green s Theorem equates the line integral of a vector field F (x, y) = P (x, y) î +Q(x, y) ĵ over with a double integral over : da. The function measures the local rotation of a 2-d field (how fast a paddlewheel at a point (x, y) will spin).

2 The Big Picture If is a closed curve in the plane (oriented counterclockwise) and is the planar region bounded by, Green s Theorem states that the line integral of a vector field F (x, y) = P (x, y) î + Q(x, y) ĵ over is equal to a double integral over, where the integrand involves partial derivatives of the components of : da. measures the local rotation of a planar vector field: how fast a paddlewheel will spin if inserted into the fields at a point (x, y). ( ) da can be thought of as the circulation of the vector field around an infinitesimal closed curve in the xy-plane: ( ) da = F dr. infinitesimal So Green s Theorem states that adding up the circulation of F around infinitesimally small curves throughout the planar region is the same as the total circulation around the boundary curve of. 2

3 How to Read Green s Theorem Given a vector field F (x, y) = P (x, y) î + Q(x, y) ĵ, Green s Theorem equates a line integral and a double integral: da Type of Integral: Line Integral ouble Integral (1-d) (2-d) omain of Integration: a closed curve in R 2 the planar region inside Integrand: F (x, y) More etails If the boundary curve is oriented in the clockwise instead of the counterclockwise direction, the values of the line integral F dr and the double integral da will have opposite signs. Geometric motivation for Green s Theorem: ( ) da measures the circulation of the field F counterclockwise around an infinitesimal curve in the plane bounding a parallelogram of area da: ( ) da = F dr. infinitesimal curve The double integral adds up these infinitesimal circulations. urves on the interior of the region share segments with their neighbors. The circulation of the field F along these shared segments have the same absolute value, but opposite signs, so they cancel. 3

4 Along the bounding curve, the infinitesimal curves have segments that don t have a neighbor to cancel with. Adding all these up leads to the total circulation counterclockwise around the bounding curve. Green s Theorem is a special case of Stokes Theorem, which we will learn about soon. Stokes Theorem relates the line integral of a 3-d field F (x, y, z) around a closed curve to the integral of the curl of F over a surface S bounded by. If F is a 2-d field and the curve lies in the plane, Stokes Theorem reduces to Green s Theorem. The integrand in Green s Theorem is the z-component of the curl of F. If F (x, y) is conservative with potential function f (x, y), then x and P as expected. = 2 f x, so by Green s Theorem: 2 f x 2 f x da = 0 da = 0 = 2 f x omparison of Green s Theorem, FTLI, FT ˆ b FT: f (b) f (a) = f (t) dt a ˆ FTLI: f (Q) f (P ) = f dr Green s: da 4

5 Notation: ifferent Ways of Writing Green s Theorem Given a vector field F (x, y) = P (x, y) î + Q(x, y) ĵ and a closed curve bounding a planar region, the following are all ways to write Green s Theorem: da da P (x, y) dx + Q(x, y) dy = da In the second equation, means the curve that is the boundary of the 2-d region. In the third equation: P (x, y) dx + Q(x, y) dy = ( ) ( ) P (x, y) î + Q(x, y) ĵ dx î + dy ĵ = F dr Using indicates that the curve is closed. can also be used. Technical onditions In order for Green s Theorem to hold, there are certain conditions that need to be met. While we do not stress these in this class, you may encounter them in future classes. Besides being closed (beginning and ending at the same point) and oriented in the counterclockwise direction, the boundary curve = needs to be piecewise smooth (made up of finitely many continuously differentiable pieces) and simple (does not intersect itself). The vector field F (x, y) needs to exist be continuously differentiable (partial derivatives of the component functions exist and are continuous) on an open neighborhood (open subset of the xy-plane) containing and its boundary curve =. This neighborhood must be simply connected (see connected for definition of simply connected ). 5

6 Here s an example of why the simply connected condition is needed. Evaluating the line integral of the field F (x, y) = y x 2 + y î + x 2 x 2 + y ĵ 2 over the unit circle x 2 + y 2 = 1 gives 2π. On the other hand, the unit circle bounds the disk, defined by x 2 +y 2 1. The partial derivatives are: x = x 2 + y 2 2x 2 (x 2 + y 2 ) 2, P = (x 2 + y 2 ) + 2y 2 (x 2 + y 2 ) 2 so x P P = 0, and the double integral over the disk is x da = 0. learly, 2π 0 = da. This is due to the fact that F (x, y) is not defined at the origin, which in contained inside the unit circle. 6