223 Homework 8

<#3#>Spring 1997<#3#>

<#5#>Due: Tuesday, May 6, 1997<#5#>

This problem will consist of three smaller problems related to the calculus of vector fields. The first problem is a classic example of flux integrals, the second is an application of the divergence theorem, and the third focuses on Stoke's Theorem.

  1. <#54#>Consider a long drainage ditch streched in the z direction and having a rectangular cross section with a width (x direction) of 30 meters and a depth (y direction) of 10 meters. Suppose there is polluted water flowing down the channel with a velocity field given by #tex2html_wrap_inline64# m/sec where the origin is at the bottom of the ditch, along the edge. We would like to remove as much of the pollutant as possible by placing a net across the ditch. Unfortunately, budget cut backs have only left us with a net that is 10 meters long in the x direction and 5 meters deep. We need to find the coordinates #tex2html_wrap_inline66# that will maximize the amount of pollution we can catch.
    1. <#12#>First place the net in the flow so that the lower right corner of the net is at the point #tex2html_wrap_inline68# and computer the flux of water through the net.<#12#>
    2. <#13#>It is natural to assume that, where the flow is fastest, there will be more pollution flowing along. We would therefore like to position the net so that the flow through it is maximal. Take the solution for the flow as a function of #tex2html_wrap_inline70# and find the coordinates which maximize the flow.<#13#>
    <#54#>
  2. <#55#>A company manufactures pollen filters for a particular device. Currently the filters produced are cubical. Someone has come up with a way to save money by making spherical filters. Your job is to find a radius for the spherical filter so that the flux of pollen through the sphere is the same as the flux through the cube. Assume the cube has a side length of c and that the pollen flows along a steady vector field given by #tex2html_wrap_inline74# Find an expression for the radius of the sphere, R, in terms of the side length of the cube.<#55#>
  3. <#61#>Consider the line integral #tex2html_wrap_inline78# of the vector field #tex2html_wrap_inline80#. Over which of the following pairs of paths and surfaces can Stokes' Theorem be applied? In each case, explain why or why not?
    1. <#56#>Let #tex2html_wrap_inline82# be the curve #tex2html_wrap_inline84# with #tex2html_wrap_inline86# and #tex2html_wrap_inline88# be the disk of radius 1 in the xy plane centered at the origin, oriented toward positive z.<#56#>
    2. <#57#>Let #tex2html_wrap_inline90# be the curve #tex2html_wrap_inline92# with #tex2html_wrap_inline94# and #tex2html_wrap_inline96# be the upper hemisphere of #tex2html_wrap_inline98# oriented downward.<#57#>
    3. <#58#>Let #tex2html_wrap_inline100# be the path #tex2html_wrap_inline102# with #tex2html_wrap_inline104# and #tex2html_wrap_inline106# is the portion of #tex2html_wrap_inline108# with #tex2html_wrap_inline110# oriented toward the positive z axis.<#58#>
    4. <#59#>Let #tex2html_wrap_inline112# be the curve #tex2html_wrap_inline114# with #tex2html_wrap_inline116# and #tex2html_wrap_inline118# be the appropriate portion of the plane z = 1.<#59#>
    5. <#60#>Let #tex2html_wrap_inline122# be the curve #tex2html_wrap_inline124# with #tex2html_wrap_inline126# and #tex2html_wrap_inline128# be the appropriate portion of the plane z = 1.<#60#>
    <#61#>