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Please help with 3 and 4.

518 Guided Projects Guided Project 77: Planimeters and vector fields Topics and skills: Vector calculus, Stokes' Theorem The planimeter is an ingenious device that allows one to trace a closed curve in the plane and determine the area of the region R enclosed by the curve (Figure 1). For this reason, it is an example of an "integrator," a mechanical device that computes areas of regions bounded by curves. The original planimeter was invented by the German engineer J.M Hermann in 1814. A popular version of the device, the polar planimeter, was invented in 1854 by Jacob Amsler. Many variations on the basic idea are manufactured today. Google offers a digital planimeter on its map sites geographical regions. measure the area of Figure The polar planimeter consists of two arms: the pivot arm OP is fixed at O and free to rotate about O, while the tracing arm PO is free to rotate about P (Figure 2). The end of the tracing arm Q is used to trace the boundary of the region of interest slides in the direction parallel to the tracing arm. As the boundary of the region is traced, the revolutions of the wheel are counted. The area of the region is proportional to the number of revolutions of the wheel. In this project we use vector fields and Stokes' Theorem to show how the polar planimeter works. wheel on the tracing arm rolls in the direction perpendicular to the tracing arm and tracing arm wheel Region pivot arm Figure 2 Planimeters and vector fields 519 1. We begin by placing the planimeter in a coordinate system (Figure 3). The pivot arm is the vector = OP = (a,b), with length u 4, the tracer arm is the vector v = PQ , with length L2, and the position vector for the end of the tracer arm is w = OQ = (x, y) = u+ v . Notice that as the boundary C of the region R is traced, the components of u, v, and w change. Show that v= (x-a,y-b). P C Region Figure 3 The wheel, which is on the vector v, always rolls in the direction orthogonal to v. Show with an explanation vector field orthogonal to v is f= (-y +b,x- a). Explain why fis determined up to a multiplicative 2. that constant. First note that fvPQL. We then have 3. f dr of Tds = cos0 ds L ds As shown in Figure 4, Tcose is the scalar projection of T in the direction of f, which is the direction in which the wheel rolls. Therefore, Tcos e ds is the distance the wheel rolls as the point Q moves along the curve a distance ds. Integrating the distances Tcos e ds over the curve C gives the total distance the wheel rolls as Q traverses C. We conclude that dr is La times the distance the wheel rolls. T|cos f Figure 4 Under suitable conditions on C, Stokes' theorem (in the xy-plane) says that f dr= 4. ndA Show that for the vector field associated with 0, we have Vxf = 2- oD Guided Projects 520 db First dy da The crux of the calculation is expressing a and b as functions of x and y, and evaluating 5. explain why a+b= \uf = Lj (x-a+(y-b)= \vP = L?' (1) (2) Differentiate (1) and (2) with respect to x and y to produce four relationships involving the partial derivatives a, a, b, and by Specifically show that 6. aa,+ bb, 0 (3) aa,+bb 0 and 7. Solve for a, and b, to show that а(у-b) b, = ау-bx ь(х— а) a, bx-ay and Combine Steps 4 and 7 to conclude that Vxf = 1 . 8. Use Step 4 and Stokes' Theorem to show that 9. Area of R L2 multiplied by the distance the wheel rolls. 10. Planimeters have a counter that registers the number of times the wheel revolves as C is traversed. Explain how the number of revolutions is converted to the area of R.

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