0
votes
0answers
10 views

How to prove that the Kronecker delta is the unique isotropic tensor of order 2?

Is there a way to prove that the Kronecker delta $\delta_{ij}$ is indeed the only isotropic second order tensor (i.e. invariant under rotation), i.e. so we can write $T_{ij} = \lambda \delta_{ij}$ ...
0
votes
1answer
19 views

Change of Eigenvalues of Ellipsoid Tensor during Rotation

I have an ellipsoid defined by the semiaxes $a,b,c$ and the orthonormal vectors $v_x, v_y, v_z$ describing the directions in which the axes point. The matrix ...
0
votes
3answers
113 views

What is the definition of rotation tensor R?

The book A First Course in Continuum Mechanics says the rotation tensor, R, is implicit in F. The matrix presentation of rotation is here. However, I am interested in its tensor representation. How ...
1
vote
3answers
132 views

What is the mathematical nature of a rotation matrix?

I have a naive question: what is the mathematical nature of a rotation matrix? Is a rotation matrix a tensor ? EDIT: if a rotation matrix is fundamentally a tensor, what is its (n, m) notation?
2
votes
1answer
55 views

What does “transform among themselves” mean?

I'm reading a script on atomic physics, and there's a chapter on irreducible tensors. I can't understand the meaning of "transform among themselves" in this context: An arbitrary rotation of the ...
8
votes
2answers
634 views

Rotation invariant tensors

It is often claimed that the only tensors invariant under the orthogonal transformations (rotations) are the Kronecker delta $\delta_{ij}$, the Levi-Civita epsilon $\epsilon_{ijk}$ and various ...
4
votes
1answer
384 views

Torque calculation, to achieve clean spin+tumble

Here's a pencil-like robotic spaceship carrying an experiment, it is a solid mass 100m long, 100 inches thick and weighs 1000kg. We're in deep solar space 100au above the sun. Assume we can apply ...
1
vote
1answer
633 views

Extracting angular velocity tensor from orthogonal matrices

Let us suppose we have two orthogonal rotation matrices representing a three-dimensional rotations $$\mathbf{R}(t)$$ and $$\mathbf{R}(t+\Delta t)$$ How is it possible to extract the angular velocity ...