Lesson 10
More on the Euler Equation

Name:faculty Section:M2 Start Time:13:26:10 Instructor:pate Course:355

1)

You probably remember that we discovered that we could use the methods of variational calculus to prove that the shortest distance between two points in two-dimensional space is a straight line (represented by the equation y = mx + b). Your everyday experience, common sense, and math classes all suggest to you that the shortest distance between two points in three-dimensional space is also a straight line (unless, of course, you want to consider a warped space-time continuum!).

As you did for the two-dimensional case, please describe how you would go about proving that this is true for the three-dimensional case, using the methods of variational calculus. Once again, don't actually do this, but describe in English the procedure you would follow. Please be specific in your steps.






2) In an end-of-summer afternoon at the beach, a friend of yours suddenly noticed a swimmer in distress out in the water. At the time he noticed the problem, he was on the beach, 400 m "downrange" of the swimmer, who was 200 m off the shore. Your friend was also 200 m from the shore line.

Make an estimate of the amount of time it might take your friend to reach the distressed swimmer. Please explain your thoughts and assumptions. (The important part of this is your thought process, not the final answer.)

Bearing in mind that your friend can run faster than he can swim, how would you determine the path he should take in order to minimize the amount of time it would take to reach the swimmer? Don't actually find the path -- just describe what you would do to determine it.




3) The "second form" of the Euler equation

  1. works when the first form does not.
  2. is only correct when the function f does not explicitly depend on x.
  3. is sometimes more convenient than the first form.
  4. more than one of the above.
  5. none of the above.



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