Integrals Over Paths and Surfaces Galileo contemplated the following question: D

Question

Integrals Over Paths and Surfaces Galileo contemplated the following question: Does a bead falling under the influence of gravity from a pont A to a point B along a curve do so in the least possible time if that curve is a circular arc? For any given path, the time of transit T is a path integral Evaluateh.fds, where f(x,y, z)=: and e(t ) = (t cos t, I sin t. 1 ) for 0 t to. 27, 28. Write the following limit as a path integral of f(x, y,z) ay over some path c on [O, 1] and evaluate: dt where the bead's velocity is v /2gy, where g is the gravitational constant. In 1697, Johann Bernoulli challenged the mathematical world to find the path in which the bead would roll from A to B in the least time. This solution would determine whether Galileo's considerations had been correct. where t...., ty is a partition of [0, 1]. 29. Consider paths that connect the points A = (0, l ) and B = ( 1,0) in the xy plane, as in Figure 7.1.5.2 (a) Calculate T for the straight-line path y = 1-r. (b) Write a formula for T for Galileo's circular path, given by (x-1)2 + (y-1 )2 = 1 Incidentally, Newton was the first to send his solution [which turned out to be a cycloid-the same curve (inverted) that we studied in Section 2.4, Example 4] but he did so anonymously. Bernoulli was not fooled. however. When he received the solution, he immediately knew its author, exclaiming, "I know the Lion from his paw." While the solution of this problem is a cycloid, it is known in the literature as the brachistrochrone. This was the beginning of the important field called the calculus of variations Circular path Path 2r figure 7.1.5 A curve joining the points Aand B. 2 Line Integrals We now consider the problem of integrating a vector field along a path. We will begin by considering the notion of work to motivate the general definition. Work Done by Force Fields If F is a force field in space, then a test particle (for example, a small unit charge in an electric force field or a unit mass in a gravitational fidwll experience the force F Suppose the particle moves along the image of a path e while being acted upon by F A fundamental concept is the work done by F on the partioa as it traces out e is a straight-line displacement given by the vector the work done by F in moving the particle along the is a con

Explanation / Answer

Velocity v is nothing but the derivative of the displacement x. So, we have

|v| = dx/dt

Now, as we know velocity is a vector quantity, meaning it has both magnitude and direction, so it can be positive as well as negative. Here, the displacement is in the negative direction. Hence the velocity v is negative and we have

v = -dx/dt

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