Harvard University,FAS
Spring 2003

Mathematics Math21a
Spring 2003

Multivariable Calculus

Course Head: Oliver knill
Office: SciCtr 434
Email: knill@math.harvard.edu
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Challenge problems

You turn in solutions to challenge problems the last day of class (Mai 1 or Mai 2). For full credit, you need 20 *'s correct. Problems *** form challenge problems. They can be hard. Problems **** form super challenge problems. They can be very hard.

Week 1: Geometry of space

  1. Topic Distances: *** Show that the set of points in the plane for which the sum of the distances from two points (-1,0) and (1,0) is constant=3 forms an ellipse: x2/a2+y2/b2=1.
  2. Topic Distances: **** An other distance in the plane is defined by d( (x,y), (u,v) ) = |x-u| + |y-v|. It is called the taxi metric or "Manhattan metric". A taxi driver in a town like Manhatten with all streets parallel to two axes experiences this distance between two points. How does an ellipse look like in this metric? You can assume that the ellipse is defined as the set of points (x,y) which have the property that the sum of the distances to (-1,0) and (0,1) is 4.
  3. Topic Distances: *** Verify that for any two vectors a and b, the inequality |a-b| | |a| - |b| | holds. Hint: Check both |a-b| |a|-|b| and |a-b| |b|-|a| using an inequality you know.
  4. Topic Dot Product: **** Given three numbers g1,g2,g3. Define a new dot product (v,w) = g1 v1 w1 + g2 v2 w2 + g3 v3 w3. For g1=g2=g3=1, this is the usual dot product.
    Which properties of the usual dot product still hold for this generalization? For which g1,g2,g3 could the dot product still serve to measure a reasonable "length" defined by |v|2 = (v,v) ?
  5. Topic: Dot Product: **** The coordinates for the edges of a cube in 4D are the 16 points +/-1, +/-1, +/-1, +/-1. Find the angle between the big diagonal connecting 1,1,1,1 with -1,-1,-1,-1 and the "middle diagonal" in one of 3D faces connecting 1,1,1,1 with -1,-1,-1,-1.
  6. Topic: Cross Product: *** Find a general formula for the volume of a tetrahedron with edges P,Q,R,S. Hint. Find first a formula for the area of one of its triangular faces, and then a formula for the distance from the fourth point to that face.

Week 2: Functions and Surfaces

  1. Topic Surfaces *** Find as many curves as you can which are obtained by intersecting two quadrics. To get full credit, you have to provide at least three different curves.
  2. Topic: Quadrics *** What surface is x2-4xy-2xz+z2=1?
  3. Topic Lines, Planes *** Two lines in the plane intersect in general in one point. Two lines in space, do not intersect in general, a line and a plane intersect in general in one point. What happens in general with the intersection of a two dimensional plane and a line in four dimensional space. What is the general intersection of two 2-dimensional planes in four dimensional space?
  4. Topic: Graphs **** How would you visualize the graph of a function f(x,y,z) of three variables? How would you describe the set {(x,y,z,u) | g(x,y,z,u)=d }, where g is a function of 4 variables and d is a constant? Take the example of the four-dimensional sphere x2+y2+z2+u2=1.

Week 3: Curves

  1. Topic Curves *** A closed curve in space is called a knot. Consider the space curve r(t)=(sin(3 t),cos(4 t),cos(5 t)). Find the smallest interval [a,b] such that this curve is a knot. Sketch the curve.
    Hint. Try first without technology. If needed, peek at here or fire up Mathematica:
  2. Topic Curves **** How could one verify that it is not possible to deform the knot r(t)=(sin(3 t),cos(4 t),cos(5 t)) into the trivial knot r(t)=(cos(t),sin(t),0) in such a way that during the deformation, the curve can never selfintersect? Hint. Look at the possible types of closed curves which don't intersect the knot. How many different types are there for the trivial knot or for the given knot? Your explanation does not need to be a precice mathematical proof but convince a general audience.

Week 4: Surfaces

  1. Topic Parametric Surfaces *** Try to graph without computer the surface r=f(theta,phi) = (2+sin(3 theta) ) (2+cos(2 phi)). (It is a graph in spherical coordinates (r,phi,theta) ). Hint: Do it in stages. First graph r=2 (the sphere), then r=(2+sin(3 theta) ), then draw a sketch of the final surface.
  2. Topic Surfaces **** In 4 dimensions, a three dimensional "surface" is described by three parameters (u,v,w). Can you find a parameterization of the three dimensional sphere x 2+y2+z2+t2=1? Hint: parameterize x2+y2=r2 with a parameter u, and z2+v2=s2 with a parameter v, then parameterize r2+s2 with the parameter w.
  3. Topic: Spherical Coordinates **** How would you design analogues of spherical or cylindrical coordinates in 4 dimensions?

Week 5: Partial Derivatives

  1. Topic Partial Derivatives ***
  2. Topic Partial Derivatives ****

Week 6: Chain Rule, Directional Derivatives

  1. Directional Derivatives *** The partial derivatives of the function f(x,y) = (x y)1/3 exists at every point but the directional derivatives in all other directions don't exist at the point (0,0). What is going on?
  2. Tangent spaces ****
    1. Extend the notion of "tangent plane" to 3-dimensional hyper-surfaces f(x,y,z,w)=c in 4-dimensional space. For example, what is the tangent plane to the three-dimensional sphere x2+y2+z2+w2=1 at the point (x,y,z,w)=(1/2,1/2,1/2,1/2).
    2. Extend the chain rule to 4 dimensions. If r(t)=(x(t),y(t),z(t),w(t)) is a curve in four dimensional space and f(x,y,z,w) is a function of four variables, what is d/dt f(r(t))?
    3. Can you use 2) to show that the gradient of f(x,y,z,w) is orthogonal to the level surface f(x,y,z,w)=const?

Week 7: Extrema

  1. Constrained extrema *** What does it mean that the Lagrange multiplier is zero in a constrained optimization problem?
  2. Extrema **** How would you do a classification of critical points for functions f(x,y,z) of three variables. Can you find a condition for a local maximum or local minimum in that case.

Week 8: Double integrals

(In both cases, the challenge is to see that this can be solved using double integrals).
  1. Double integral *** Integrate arccos(x1/2) over the interval [0,1].
  2. Double integral **** Integrate the function exp(-x2) over the real line.

Week 9: Triple Integrals

  1. Volume *** Find the volume of the intersection of two cylinders y2+z2=1 and x2+z2=1. You have a picture on page 882 in the text.
    Hint. Look what happens, when you cut the body at a fixed z value and calculate the area of this section.
  2. Quadrupel integral **** Use a quadruple integral to find the volume of the hyper-sphere x2+y2+z2 +w2=r2 in four dimensional space. Hint. If you slice up the hyper-sphere at w, you get a sphere of radius (r2-w2)1/2. Integrate the volume of the sphere from w=-r to w=r using substitution.

Week 10: Line Integrals

1) Line integrals **** Consider a O shaped pipe which is filled only on the right side with water. A wooden ball falls on the right hand side in the air and moves up in the water. Why does this "perpetuum mobile" not work?

2) Line integrals **** What is wrong with the Escher picture to the left which describes water always falling down? The figure suggests the existence of a force field which is not conservative, a perpetual motion machine.

Week 11: Integral theorems I

  1. Line integrals *** What part of the theory of line integral still holds, when working with a more general dot product v . w = g1 v1 w1 + g2 v2 w2 + g3 v3 w3? How could we save the fundamental theorem of line integrals or Green's theorem? (Hint: Think about modifying the definition of the gradient).
  2. Vectorfields Div Grad **** An electric charge has a field F which is the gradient of f(x,y) = 1/|(x,y)|. Assume you have three positive electric charges located on a equilateral triangle with side length 1.
    a) Without writing down any formula, sketch the field F in the plane generated by the three charges. The field is the sum of the fields of the three charges.
    b) What is curl(F) for points (x,y) away from the location of the charges? (Hint: look at one charge alone first).
    c) What is div(F) for points (x,y) away from the location of the charges? (Hint: look at one charge first).
    d) Does Green's theorem hold true for all regions G in the plane?
    (Hint: Again look first at one charge alone and consider the case of a circular region centered at that charge.)

Week 12: Integral theorems II

  1. Divergence theorem *** Consider a hollow sphere of outer radius R2 and inner radius R1. Compute the gravitational field in dependence of the distance to the center of that "hollow world".
  2. Curl, Conservative fields **** Nash's problem:
    Find a subset X of R3 with the property that if V is the set of vector fields F on R3 - X which satisfy curl(F)=0 and W is the set of vector fields F which are conservative: F= f. Then, the space V/W should be 8 dimensional.

    Remark. The meaning of the last sentence is that there should be 8 vectorfields Fi which are not gradient fields and which have vanishing curl outside X. Furthermore, you should not be able to write any of the 8 vectorfields as a sum of multiples of the other 7 vector fields.

Please send comments to math21a@fas.harvard.edu

Wed May 21 22:22:43 EDT 2003