2025年1月26日 · The tensor product of two 1 dimensional vector spaces is 1 dimensional so it is smaller not bigger than the direct sum. The tensor product tof two 2 dimensional vector spaces is 4 dimensional so this is the the same size as the direct sum not bigger.This is correct but missing the relevant point: that the presentation contains a false statement.

A tensor field of type $(0, 0)$ is a smooth function. A tensor field of type $(1, 0)$ is a vector field. A tensor field of type $(0, 1)$ is a differential $1$-form. A tensor field of type $(1, 1)$ is a morphism of vector fields. A tensor field of type $(0, 2)$ which is symmetric and nondegenerate is a metric tensor.

2014年1月30日 · The complete stress tensor, $\sigma$, tells us the total force a surface with unit area facing any direction will experience. Once we fix the direction, we get the traction vector from the stress tensor, or, I do not mean literally though, …

2013年6月6日 · The components of a rank-2 tensor can be written in a matrix. The tensor is not that matrix, because different types of tensors can correspond to the same matrix. The differences between those tensor types are uncovered by the basis transformations (hence the physicist's definition: "A tensor is what transforms like a tensor").

2005年7月14日 · where [itex]\sigma[/itex] is the Cauchy (true) stress tensor. The symmetricity of the Cauchy stress tensor arises from the law of balance of moment of momentum (complete presentations, and best IMO, are typically in thermomechanics books & papers), [tex] \oint (r\times T) dA + \int (r \times \rho b) dV = \frac{D}{Dt} \int (r \times \rho v) dV ...

2011年10月25日 · It's in a book called "Vectors, tensors, and the basic equations of fluid mechanics" by Rutherford Aris, and method can be found in chap 2.7. The basic idea is to first show Levi-Civita is indeed invariant, second by considering a few special rotations to show that if a rank-3 3-D tensor has to be Levi-Civita before it can be an isotropic tensor.

A tensor field has to do with the notion of a tensor varying from point to point . A scalar is a tensor of order or rank zero , and a scalar field is a tensor field of order zero . A vector is a tensor of order or rank one , and a vector field is a tensor field of order one . …

2017年5月31日 · Consider the following example. Let's say we have a certain vector v that we would like to first transform somehow and then rotate.(Lets say that the transformation we want to apply to v is a simple scaling, which is simpler to describe wrt to orthodox v i.e. before we rotate it) This leads us to having two options for the operations:

2011年1月25日 · Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology and algebraic geometry. and tensor products of vector spaces are also important in differential geometry and physics. I think it is better to learn about these applications thoroughly than to have someone attempt to summarize them.

Tensor Product of Algebras: Multiplication Definition. 9. Alternative introduction to tensor products of ...