Solutions exam 2 - 2.035 Selected Topics in Mathematics...

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2.035: Selected Topics in Mathematics with Applications Final Exam Spring 2007 Every problem in the calculus of variations has a solution, provided the word “solution” is suitably understood. David Hilbert (1862-1943) Work any 5 problems. Pick-up exam: 12:30 PM on Tuesday May 8, 2007 Turn-in solutions: 11:00 AM on Tuesday May 15, 2007 You may use notes in your own handwriting (taken during and/or after class) and all handouts (including anything I emailed to you) and my bound notes. Do not use any other sources. Do not spend more than 2 hours on any one problem. Please include, on the first page of your solutions, a signed statement confirming that you adhered to all of the instruction above. 1
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Problem 1: Using first principles (i.e. don’t use some formula like d/dx ( ∂F/∂φ ) ∂F/∂φ = 0 but rather go through the steps of calculating δF and simplifying it etc.) determine the function φ A that minimizes the functional 1 1 F { φ } = ( φ ) 2 + φφ �� + φ dx 2 0 over the set A = { φ | φ : [0 , 1] R , φ C 2 [0 , 1] } . Note that the values of φ are not specified at either end. ——————————————————————————————————————– Problem 2: A problem of some importance involves navigation through a network of sensors. Suppose that the sensors are located at fixed positions and that one wishes to navigate in such a way that the navigating observer has minimal exposure to the sensors. Consider the following simple case of such a problem. See figure on last page. A single sensor is located at the origin of the x, y -plane and one wishes to navigate from the point A = ( a, 0) to the point B = ( b cos β, b sin β ). Let ( x ( t ) , y ( t )) denote the location of the observer at time t so that the travel path is described parametrically by x = x ( t ) , y = y ( t ) , 0 t T . The exposure of the observer to the sensor is characterized by T E { x ( t ) , y ( t ) } = I ( x ( t ) , y ( t )) v ( t ) dt 0 where v ( t ) is the speed of the observer and the “sensitivity” I is given by 1 I ( x, y ) = . x 2 + y 2 Determine the path from A to B that minimizes the exposure. Hint: Work in polar coordinates r ( t ) , θ ( t ) and find the path in the form r = r ( θ ). Remark: For further background on this problem including generalization to n sensors, see the paper Qingfeng Huang, ”Solving an Open Sensor Exposure Problem using Variational Calculus”, Technical Report WUCS-03-1, Washington University, Department of Computer Science and Engineering, St. Louis, Missouri, 2003. ——————————————————————————————————————– Problem 3: Consider a domain D of the x, y -plane whose boundary D is smooth. Let A denote the set of all functions φ ( x, y ) that are defined and suitably smooth on D and which vanish on the boundary of D : φ = 0 for ( x, y ) D . Define the functional 1 2 φ 2 1 F { φ } = D 2 ∂x∂y + 2 φ 2 + dA 2
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for all φ A where q = q ( x, y ) is a given
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