Honors Calculus 252-2

Problem Set 4

to be handed in Friday February 8, 2002, in class

1. A bug is crawling radially outward at a speed C on a record player rotating at an angular velocity W.
   1. Show that its velocity at a point (x,y) is given by
      

      v(x,y) = æ ç ç ç è - Wy + Cx   _____ Öx2+y2 , Wx + Cy   _____ Öx2+y2 ö ÷ ÷ ÷ ø

   2. If the bug is initially at a distance D from the center and at an angle a show that its polar coordinates after a time t are
      

      r(t) = D+Ct        q(t) = a+ Wt .

      Compute its cartesian coordinates x(t),y(t).

   3. Show that the bug's trajectory in (b) is a streamline of the vector field v.

2. The electrostatic potential, F, is defined as E = - ÑF. Consider an electric field,
   

   E = R e- R2 ^ R   ,

   where [^R] is the unit vector in the radial direction in a spherical coordinate system. Construct F(x,y,z) by doing a line integral along a radial path starting at the origin; assume F(0,0,0) = 0. Verify your result by evaluating ÑF.

3. 8.1.5, 8.1.8(i)

4. 8.2.5, 8.2.14(i), 8.2.14(ii)