Math 311 Spring 2002: Homework problems


  •   1    Jan 14  -    Sections 1.1-1.7 :  Vector algebra. Geometry.  S 1.5: 3,4*,5,9,12,13,16*,   S 1.7: 8*,9*
  •   2    Jan 16  -    Section  1.8          :   Equations of lines.   S 1.7: 14,16-19,20*,21,22*,23-24,   S 1.8: 1,3,4,8
  •   3    Jan 18  -    Sections 1.9,1.10 :   Angles. Dot Product. Equations of planes.
  •                                                             S 1.7: 2
  •                                                             S 1.8: 11* (first show that the lines intersect, then find the angle between them),18
  •                                                             S 1.9: 4,7*,11*,12a
  •                                                             S 1.10: 1*,2,3,4*,6,7a,8*,9,12*,13*

    • MARTIN LUTHER KING DAY
  •   4    Jan 23  -    Sections 1.12- 1.14 : Cross Product. Triple product. Vector Identities.
  •                                                          S 1.12: 3*,5,8*,9*,15*(find the line of instersection of the two planes,
  •                                                                      not a parallel line as stated in the book),19*,22
  •                                                          S 1.13: 8*
  •                                                          99*: Prove that [a,b,c]=[b,c,a], and that [b,c,a]=[c,a,b]
  •                                                          S 1.14: 1*
  •   5    Jan 25  - Sections 2.1-2.2: Space curves. Velocity, tangent vectors, smooth curves, arclength.
  •                                                          100*: Prove that [x(t).y(t)]'=x'(t).y(t) + x(t).y'(t) where x(t),y(t)
  •                                                                       are differentiable vector valued functions of one variable
  •                                                           101*: Show that x(t)=<t^2,t^3>, -1<=t<=1 is not a smooth curve,
  •                                                                       since it has a corner in it (Hint: find a nonparametric repre-
  •                                                                       sentation of the curve), even though the two components t^2
  •                                                                       and t^3 are continuously differentiable functions of t. Why
  •                                                                       does this not contradict the
  •                                                                       definition of a smooth curve that we gave in class?
  •                                                            S 2.1:4*
  •                                                            S 2.2:1,2,3,4,5*,7,8*

  •   6    Jan 28  - Section 2.3: arclength, curvature, acceleration. No Homework.
  •   7    Jan 30  - Section 2.3: acceleration, torsion, Frenet formulas.
  •                                              S 2.3: 4,5,6,9*,10*(also find curvature),13,15*,17*
  •                                              102*: (a) Prove that dT/ds is normal to T  (we did this in class)
  •                                                        The vector B is defined by B=TxN
  •                                                        (b) Prove that dB/ds is normal to B
  •                                                        (c) Prove that dB/ds is normal to T
  •                                                        (d) Deduce that dB/ds has to be parallel to N, that is, dB/ds = alfa*N
  •                                                               we call  tau=-alfa  the torsion of the curve
  •                                                        (e) Use the result in (d) and the fact that dT/ds=kappa*N to show that
  •                                                               dN/ds = -kappa*T + tau*B  (Hint: start with the fact that N=BxT)
  •                                                         This completes the derivation of the Frenet formulas.
  •   8    Feb  1  - Section 2.4: motion in polar coordinates.  S 2.4: 1*,3,4*,8,13*,14*
  •                                              103*: Use Equation 2.51 to find the acceleration of a particle moving around a circle
  •                                                        centered at the origin with constant angular velocity w. Compare with the
  •                                                         result of Example 2.16.
  •   9    Feb  4   - Section 3.1: Scalar Fields. Gradients.   S 3.1: 3,5*,8,9*,11,12,14,16*,22*,23
  • 10    Feb  6  -  Section 3.2,3.3: Vector Fields. Divergence.   S 3.2: 3,4,    Section 3.3: 2,3*,4,5*,7*,8*,11, skip 12 (much too messy)
  • 11    Feb  8  -  Section 3.4: Curl.  S3.4: 4,7,9   S 3.5: 4*,5*,6*,7,9*,10
  •                                                  104*: Graph the following vector fields (by drawing a representative
  •                                                              set of arrows) and compute the curl in each case:
  •                                                               (a)  F(x,y) = <-y,x>      (b) F(x,y)=<-y,x>/(x^2+y^2)
  •                                                               Conclude that the fact that a particle is rotating about an axis
  •                                                               does not imply that the curl is nonzero at that particle.

  • 12    Feb 11  - Section 3.6: Laplacian.  S 3.6:  2,3,4,5*,6a*,7*
  • 13    Feb 13  - Section 3.8: Vector Identities.  S 3.8: 1*,3,10*,13*
  • 14    Feb 15  - **EXAM 1**

  • 15    Feb 18  - Section 3.10: Cylindrical coordinates.   No Homework.
  • 16    Feb 20  - DUE MONDAY 2/25:
  •                          Section 3.10: Cylindrical coordinates ctd.   S 3.10: 2*,3*,6*,7*,8*,10*,11*,12*
  •                          105*:  Find formulas for e_rho, e_phi, e_theta in terms of rho,phi,theta.
  •                                     (Hint: start with Pb 4 in S3.10)
  •                           106*: Find the derivatives of e_rho, e_phi, e_theta in terms of these vectors(see class notes)
  •                           Extra credit : Find div.F in spherical coordinates using the results of 105,106.
  • 17    Feb 22  - Section 3.10: Spherical coordinates. No Homework.

  • 18    Feb 25  - Section 4.1: Line Integrals.  S 4.1: 1,2,3*,4*,7*,8*,10,14*,19*
  • 19    Feb 27  - Section 4.3:  Conservative Fields.   S 4.3: 1*,2(a*,b,c),3(a*,b,c),4*,5*,6,7*
  • 20    Mar  1  - Sections 4.3,4.4:  Conservative Fields.  S 4.4: 1(a*,b,c*,d,e*),2*,3*,4,5,6,9b*,10

  • 21    Mar  4  - Section 4.5: Solenoidal Fields (incompressible).  HW #8, due Fri  3/8: S 4.5: 2*,4,9*,10*
  •                          107*: Show that if F=<F_1,F_2,0> is a 2D incompressible field then the line integral
  •                                    chi=int_(x_o,y_o)^(x,y) -F_2 dx + F_1dy is path-independent (and hence it
  •                                    is well defined without specifying path)
  • 22    Mar  6  - Section 4.6: Surfaces: Orientation and Parametrization.   HW #8, due Fri 3/8  S 4.6: 7*,
  •                                    108*: Find  (a) parametric, (b) nonparametric equations for the plane spanned
  •                                     by a=<2,1,0>  b=<3,-1,1> through the point P(2,2,0)
  • 23    Mar  8  - Section 4.6: Surface element, surface areas.   No Homework.

    • SPRING BREAK
  • 24    Mar 18  - Section 4.6: Surface areas.  HW#9, due Mon 3/22:  S 4.6 1,2,3*,4*,5,8,109*:Find the surface
  •                                               area of the saddle f(x,y) = x^2-y^2 for  x^2+y^2<=1   (Hint: to evaluate the
  •                                               integral, change to polar coordinates, then use the tables in the back of your
  •                                              calculus book)
  • 25    Mar 20  - Quiz #2 (long quiz on HW #7 and #8, no lecture)
  • 26    Mar 22  - Section 4.7: Surface Integrals.

  • 27    Mar 25  - Section 4.7: Surface integrals. HW#10, due Fri 3/29:   S 4.7: 1,2(a,d,f),3,5*,7*,10,11*,
  •                                                110*: Evaluate the integral over S of  x^2yz dS where S is the surface given by z=1+2x+3y
  •                                                           and  x is in [0,3] and y is in [0,2]
  •                                                111*: Evaluate the integral over S of F.ndS where F=<x,y,z> , S is the sphere x^2+y^2+z^2=9
  •                                                          and n is the inward normal
  •                                                112*: Evaluate the integral over S of F.ndS where F=<-y,x,3z> ,  S is the hemisphere
  •                                                           x^2+y^2+z^2=16, z>=0, and n is the upward normal
  • 28    Mar 27  - Gauss Law. Volume integrals. HW#10, due Fri 3/29:   S 4.7: 18*
  •                                                113*: Evaluate the integral over V of xz dV where V is the tetrahedron with vertices
  •                                                          (0,0,0), (0,1,0), (1,1,0), (0,1,1)
  • 29    Mar 29  - Volume Integrals.  HW #11, due Fri 4/5:  S 4.8: 1,3,4*,5*,6
  •                                                114*: Sketch the region V bounded by 4x^2+z^2=4, y=0 and y=z+2. Find the volume of V.
  •                                                115*: Sketch the region V bounded by x=y^2, z=0 and x+z=1. Find the volume of V.

  • 30    Apr   1  -  Divergence Theorem.   HW#11, due Fri 4/5:   S 4.9:  1,2(a,d,f),3,4*,5*,6,
  •                                                116*: Verify the Divergence Theorem for F=<-y,x,3z> and S:x^2+y^2+z^2=16, z>=0.
  •                                                          You may use results from Problem 112.
  •                                                117*: Use the Divergence Theorem to evaluate the integral over S of F.n dS where S is
  •                                                           the spherical surface x^2+y^2+z^2=9, n is the outward normal, and F=<x,y,z>.
  •                                                           Compare with your result in Problem 111.
  • 31    Apr   3  -  Divergence Theorem.  HW#11, due Fri 4/5:   S 5.1:  6*,7*, 8*,9*,10*
  • 32    Apr   5  -  Review for exam. Divergence Theorem for 2D fields: Green's Theorem. HW: Review for exam.

  • 33    Apr   8  -  Stokes Theorem.  HW#12, due mon:   S 4.9:  7*,8,9,10*,11,12*,14*,16
  • 34    Apr 10  -  Stokes Theorem.  HW#12, due mon:   S 4.9:  17*,18*,23,26*,   S 5.5:  1*,2*,5a,6
  • 35    Apr 12  -  EXAM #2  (Lectures 15-31)

  • 36    Apr 15  - Greens Formulas. No Homework.
  • 37   Apr 17  - Greens Formulas and the Poisson equation.  HW#13, due mon:   S 5.2: 1*,2*,5*,
  •                                               118*: Derive the 1st and 2nd Green's identities.
  • 38    Apr 19  - Fundamental Theorem of Vector Analysis.  HW#13, due mon:   S 5.3:  3*,4*,7*.

  • 39    Apr 22  - Fundamental Theorem of Vector Analysis.   No Homework.
  • 40    Apr 24  - Sources, Sinks, Vortex lines. Applications to Electrostatics and Fluids.
  • 41    Apr 26  - Applications: Electrostatics (Gauss Theorem), Conservation of Mass.

  • 42-44    Apr 29-May 3  -   REVIEW WEEK


    •   FINAL EXAM,   Wednesday, May 8, 7:30-9:30am   (in regular classroom)