Math 603-601 - Fall 2002
Homework
Assignment 8
- 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.
- Let B = {f1 = i+j+k,
f2 = i-j+k,
f3 = j-k}.
- Find the metric tensor g. Use it to find the basis reciprocal to
B, B* = {f1, f2,
f3}.
- 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*.
- Verifiy that all of the following are the same:
u·v,
[v]BTg[u]B,
[v]B*Tg-1[u]B*,
[v]B*T[u]B, and
[v]BT[u]B*
- 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 uk and vk 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.
- 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.)
- 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 Aj be the area of the face with outward
normal -ej, where e1 = i,
e2 = j, and e3 = k.
- 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|.)
- Show that the volume V of the tetrahedron is given by V =
(2A1A2A3)½/3 and
that V < (2½/3)A3/2.
Due Thursday, 14 November.