Math 603-601 - Fall 2002

Assignment 8

1. Find the solution to the system dx/dt = Ax, if x(0) = [1 1]^T and A =

    3  4
   -1 -1
   

   Note: A is not diagonalizable. You need to find a basis for which A is in Jordan normal form.

2. Let B = {f₁ = i+j+k, f₂ = i-j+k, f₃ = j-k}.
   1. Find the metric tensor g. Use it to find the basis reciprocal to B, B^* = {f¹, f², f³}.
   2. For each of the vectors u = 3i-j-2k and v = i-4j+3k, find the contravariant and covariant components relative to B and B^*.
   3. Verifiy that all of the following are the same: u·v,  [v]_B^Tg[u]_B,  [v]_B*^Tg^-1[u]_B*,   [v]_B*^T[u]_B, and [v]_B^T[u]_B*

3. This exercise concerns the cross product. Let B = {f₁, f₂, f₃} be a basis for 3D displacements, and let B^* = {f¹, f², f³} be the basis reciprocal to B. In addition, let W = f₁·f₂×f₃. be the "signed" volume of the parallepiped formed by the vectors in B. If u^k and v^k are the kth contravariant coordinates for u and v, respectively, then show that u×v is the determinant of

   Wf1	Wf2	Wf3
   u1	u2	u3
   v1	v2	v3

   Show that if B = {i,j,k}, then this reduces to the familiar form of the cross product.

4. Find the inertia tensor about the center of mass for a rigid body composed of eight equal masses placed at the vertices of a cube centered at the origin and having sides of common length 2a. (That is, the eight vertices are (a,a,a), (a,a,-a), (a,-a,a), and so on.)

5. Consider the tetrahedron with vertices (0,0,0), (a,0,0), (0,b,0), (0,0,c). Let A and n be the area and normal for the inclined face, and let A_j be the area of the face with outward normal -e_j, where e₁ = i, e₂ = j, and e₃ = k.
   1. Show that A n = A₁e₁ + A₂e₂ + A₃e₃. (Hint: the area of a triangle in space formed by vectors U and V is ½|U×V|.)
   2. Show that the volume V of the tetrahedron is given by V = (2A₁A₂A₃)^½/3 and that V < (2^½/3)A^3/2.