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Unformatted text preview: 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 (18621943) Work any 5 problems. Pickup exam: 12:30 PM on Tuesday May 8, 2007 Turnin 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 Problem 1: Using first principles (i.e. don’t use some formula like d/dx ( ∂F/∂φ ) − ∂F/∂φ = 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 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...
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This note was uploaded on 02/24/2012 for the course MECHANICAL 2.035 taught by Professor Rohanabeyaratne during the Spring '07 term at MIT.
 Spring '07
 RohanAbeyaratne

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