Thursday March 6 Solutions Integrating vector fields 1 Consider the vector field F y 0 on R2 a Draw a sketch of F on the region where 2 x 2 and 2 y 2 Check your answer with the instructor SOLUTION Below is the image for parts a and b b Consider the following two curves which start at A 2 0 and end at B 2 0 namely the line segment C 1 and upper semicircle C 2 R R Add these curves to your sketch and compute both C 1 F d r and C 2 F d r Check you answers with the instructor SOLUTION Parametrize C 1 by r1 t t 0 0 t 2 and parametrize C 2 by r2 t 2 cos t 2 sin t 0 t We have Z C1 Z C2 Z F dr 0 2 Z F dr 0 F r1 t r01 t d t F r2 t r02 t d t Z 0 Z 0 2 0 0 1 0 d t 0 Z 2 sin t 0 2 sin t 2 cos t d t 4 sin2 t d t 0 1 1 t sin 2t 2 4 2 2 0 c Based on your answer in b could F be f for some f R2 R Explain why or why not SOLUTION R By the Fundamental Theorem of Line Integrals if F f for some f R2 R then C F d r is path independent for any curve C starting at A 2 0 and ending at B 2 0 Since we obtained different answers for the paths C 1 and C 2 F cannot be of this form 2 Consider the curve C and vector field F shown below a Calculate F T where here T is the unit tangent vector along C Without parameterizing C evaluate R R C F d r by using the fact that it is equal to C F T ds SOLUTION 3 From the picture we suppose that F x y 1 1 We have T p1 2 1 so F T p So 5 5 Z Z Z 3 F d r F T ds p ds 3 C C 5 C R since C ds is simply the distance between 1 1 and 3 2 b Find a parameterization of C and a formula for F Use them to check your answer in a by comR puting C F d r explicitly SOLUTION Parametrize C by r t 3 2t 2 t 0 t 1 We already have F 1 1 So Z Z C F dr 0 1 1 1 2 1 d t 3 3 Consider the points A 0 0 and B 2 Suppose an object of mass m moves from A to B and experiences the constant force F mg j where g is the gravitational constant a If the object follows the straight line from A to B calculate the work W done by gravity using the formula from the first week of class SOLUTION Recall that the work done on an object moving along a straight line subject to a constant force F is W F D where D is the displacement vector In this case D 2 and F 0 mg So W 2 0 mg 2mg b Now suppose the object follows half of an inverted cycloid C as shown below Explicitly parameterize C and use that to calculate the work done via a line integral SOLUTION A parametrization for the inverted cycloid C is r t t sin t cos t 1 0 t So Z W C Z F dr 0 Z 0 mg 1 cos t sin t d t 0 mg sin t d t mg cos t 0 2mg c Find a function f R2 R so that f F Use the Fundamental Theorem of Line Integrals to check your answers for a and b Have you seen the quantity f anywhere before If so what was its name SOLUTION If such an f exists we must have f x 0 and f y mg Integrating mg with respect to y we obtain f mg y C x where C x is some function of x Differentiating this with respect to x we obtain f x C 0 x 0 so f mg y K where K is a constant is a potential function for F By the Fundamental Theorem of Line integrals both a and b must have the same answer namely Z Z L F dr L f d r f B f A f 2 f 0 0 mg 2 K K 2mg where L is the line segment from A to B and Z C Z F dr C f d r f B f A 2mg The quantity f is called the potential energy 4 If you get this far work 52 from Section 16 2 SOLUTION We are assuming that B has magnitude which only depends on the distance from the wire So B B is constant along any circle centered around the wire in a plane perpendicular to the wire Let C r t be such a circle with radius r parametrized in the counterclockwise direction and let B denote the magnitude of B along C Note that B r t is a positive multiple of r0 t by definition So it follows that B r t T t the unit tangent vector to C is given by T t We have B Z Z Z Z B B B d r B Tds ds B ds 2 r B C C C B C By Ampere s Law R C B d r 0 I so we have 2 r B 0 I or B 0 I 2 r
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