Tensor Analysis: quadric, bases, MOI, bisectors

[domath]

Homework Statement

1) Let e, e' be two orthonormal bases(not necessarily the standard basis). Prove that the matrix S = [Id]e-e' (direct transformation from e to e') is orthogonal, i.e. SS^T = S^TS = [Id]_3. SS^T is S * S transpose.

2) The moment of inertia tensor of a right circular cylinder with respect to the axes passing through its center of mass(the x_3-axis is parallel to the generators) is of the form

||I_ik|| = ||I_0, 0, 0; 0, I_0, 0; 0, 0, I_1||

Find the moment of inertia of the cylinder about the bisectors of the angles between the various coordinate axes.

3) Prove that the matrix a_ij is a rank 2 tensor, where Q is a quadric given in Euclidean coordinates by the equation

summation_ij (a_ij * x_i * x_j) = 1

Homework Equations

The Attempt at a Solution

1) S = S reciprocal? [Id]e-e' = (e'^T)*(e)
2) No clue.
3) More info needed?

 

[mighty2000]

1) To prove that the matrix S is orthogonal, we need to show that SS^T = S^TS = [Id]_3.
First, let's calculate SS^T:
SS^T = [Id]e-e' * [Id]e-e'^T
= (e'^T)*(e) * (e'^T)*(e)
= (e'^T)*(e')
= [Id]_3
Therefore, SS^T = [Id]_3.

Next, let's calculate S^TS:
S^TS = [Id]e-e'^T * [Id]e-e'
= (e)*(e'^T) * (e)*(e'^T)
= (e)*(e')
= [Id]_3
Therefore, S^TS = [Id]_3.

Since both SS^T and S^TS equal [Id]_3, we can conclude that S is an orthogonal matrix.

2) To find the moment of inertia of the cylinder about the bisectors of the angles between the various coordinate axes, we can use the parallel axis theorem. This theorem states that the moment of inertia about a parallel axis is equal to the moment of inertia about the center of mass plus the product of the mass and the square of the distance between the two axes.

In this case, the moment of inertia about the bisectors of the angles between the various coordinate axes will be equal to the moment of inertia about the center of mass (which is given in the problem as ||I_0, 0, 0; 0, I_0, 0; 0, 0, I_1||) plus the product of the mass and the square of the distance between the two axes.

Since the distance between the two axes is equal to the radius of the cylinder, we can write the moment of inertia as:
||I_ik|| = ||I_0, 0, 0; 0, I_0, 0; 0, 0, I_1|| + mr^2

3) To prove that the matrix a_ij is a rank 2 tensor, we first need to understand what a rank 2 tensor is. A rank 2 tensor is a linear transformation that takes in two vectors and produces a scalar. In this case, the tensor a_ij takes in two vectors x_i