MAS 4156/5157 Homework Assignments
Spring 1999

Homework problems and approximate due dates are listed below. Read the corresponding section of the book before working the problems.

  • Section 1.2: 1-4 (due 1/8/99).
  • Section 1.3: 1-3, 5-11, 14. In #14, assume A and B are nonzero. Hint: use unit vectors. (due 1/8/99)
  • Section 1.4: 1-7, 9c, 13 (note: ``two solutions'' means ``two answers'', not ``two methods of finding one answer) (due 1/8/99).
  • Section 1.5: 1-10 (due 1/11/99).
  • Section 1.6: 1-5, 7 (due 1/11/99).
  • Section 1.7: 1-4, 11, 13-16 (due 1/13/99), 18-27 (due 1/15/99). (The problems from #13 on are intended to help you review your 3-dimensional analytic geometry. If you have trouble with these problems, you need to start reviewing in more depth NOW so that you are ready to use these tools when we get to chapters 4 and 5.)
  • Section 1.8: 1-10, 15, 18 (due 1/13/99).
  • Section 1.9: 1,3,4,6,7,9,11-14,16-18, 22, 26 (due 1/13/99)
  • Section 1.10: 1-7, 14 (see Example 1.23), 20, 22, 24 (due 1/15/99).
  • Section 1.12: 1,3-10, 13, 18-21, 28 (due 1/20/99)
  • Section 1.13: 1,3,6,8,14,17 (note: this is a version of Cramer's Rule for solving 3 equations in 3 unknowns) (due 1/22/99)
  • Section 1.14: 1-3, 6, 7, 9 (note: there are quick ways of doing 1-3) (due 1/22/99)
  • Section 2.1: 1-4 (due 1/27/99)
  • Section 2.2: 1-8, 12 (note: ``unit pitch'' means ``pitch=1''), 13-15 (due 1/27-29/99).
  • Section 2.3: 1-6, 11 (due 1/29/99)
  • Section 2.3 continued: Read the remainder of section 2.3 by 2/3/99, and do the following problems by 2/5/99: 15, 17, 18, 20. Also, due the class meeting after we discuss evolutes are the following problems: (A) Find and sketch the evolute of the ellipse in the xy plane whose equation is x 2 /4 + y 2 =1. (B) show that the evolute of a helix is another helix with the same axis, same handedness, and same curvature, but possibly different torsion.
  • Section 2.4: 1,3,6, 16 (due 2/12/99)
  • Section 3.1: 1,3,4,6,8a,10,14,15,19,20,21, 23, 24, 28, 31, 32 (due 2/17/99)
  • Section 3.2: 1,3,4 (due 2/19/99)
  • Section 3.3: 1-5, 7, 8, 10. Also do problems (A), (B), and (C) below. In these problems, r stands for distance to the origin in 3-space, i.e. (x2+y2+z2)1/2, and ur denotes the unit vector field R/|R|=R/r =(xi+yj+zk)/r, defined everywhere except at the origin. A scalar function f defined on 3-space except possibly at the origin is called radially symmetric if f(x,y,z)=g(r) for some scalar function g of one variable. A vector field F defined everywhere in 3-space except possibly at the origin is called radially symmetric if at all points other than the origin, F(R)=g(r)ur for some scalar function g. (For example, taking g to be the constant function 1, ur is itself a radially symmetric vector field. Another example is the inverse-square radial vector field -r-2ur.) (A) Show that the gradient of a differentiable radially symmetric function is a radially symmetric vector field. (B) Show that the divergence of a radially symmetric vector field is a radially symmetric function. (C) Using problem (A) and the book's problem 5, show that if F is a vector field everywhere orthogonal to ur and if div(F)=0 everywhere, then for every differentiable radially symmetric function g, div(gF)=0 everywhere. (In this problem "everywhere" means "at every point except possibly the origin".) (due 2/22/99)
  • Section 3.4: 1,4-12. Before doing #12, do this problem: show that if g is a scalar function and F is a vector field, then curl(gF)=(grad g) x F +g curl(F). (Here x means cross product.) Note that the result of #12 can be rephrased as "the curl of a radially symmetric vector field is identically zero". (due 2/24/99)
  • Section 3.5: 1-8 (due 2/26/99)
  • Section 3.6: 1-5, 7, 8. In #4, recall that the functions cosh and sinh are defined by cosh(x)=(ex+e-x)/2, sinh(x)= (ex-e-x)/2. Note that each of these functions is the derivative of the other (without any sign changes as in cos and sin). (due 2/26/99)
  • Section 3.8: 2, 3 (eq. (3.42) is the only one we did not do in class) 5, 7 (due 3/1/99)
  • Section 3.9: 1, 2, 4 (due 3/1/99)
  • NOTE DATE CHANGE FOR SECOND EXAM. NEW DATE IS WEDNESDAY 3/3/99.
  • Section 4.1: Read examples 4.2-4.3 and do problems 1-10, 20 (due 3/15-17/99)
  • Section 4.2: 1-10 (In #2, the region is the set of all points in space other than the locations of the n point charges) (due 3/17/99)
  • Section 4.3: 2ace, 3ace, 4-8 (due 3/19/99)
  • Section 4.4: 1-6, 8a, 9, 10 (hint for 6bc: after writing out the t-integrals and seeing how horrible they are, find a potential for F and use Theorem 4.1 instead of doing the t-integrals directly) (due 3/22/99)
  • Section 4.5: Read from the beginning of the section up through Example 4.10 and do problems 1 and 2. In problem 2, it should say a vector potential, not the vector potential. (due 3/24/99)
  • Section 4.6: 1-5, 8, 10 (due 3/29/99)
  • Section 4.7: 1, 2cde, 5, 6, 7, 11, 13,15, 16 (due 3/29-31/99)
  • NOTE DATE CHANGE FOR THIRD EXAM. NEW DATE IS FRIDAY 4/2/99.
  • Section 4.8: Read section and do problems 3, 4, 6 (due 4/5/99)
  • Section 4.9: 1, 2cde, 3b, 6 (hint: see problem (B) in the homework above for section 3.3), 17, 20 (due 4/9/99)
  • Section 4.9 continued: 7, 8, 9b, 10-12, 16 (due 4/12/99; more problems may be announced and due later)
  • Section 5.1: 7-9 (due 4/12/99)
  • Section 5.4: 1-4, 6, 8, 9 (due 4/16/99)
  • Section 5.5: 1-3 (due 4/16/99)
  • Section 4.9 continued: 23, 31 (due 4/16/99)
  • Section 5.2: 1, 2 (due 4/19/99)
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    This page was last modified by D. Groisser on Apr. 17, 1999.