In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry , algebraic geometry , general relativity , in the analysis of stress and strain in material object, and ...

在数学，物理和工程上， 张量场 （tensor field）是一个的非常一般化的几何变量的概念。 它被用在 微分几何 和流形的理论中，在代数几何中，在广义相对论中，在材料的应力和应变的分析中，和在物理科学和工程的无数应用中。

Tensors and transformations are inseparable. To put it succinctly, tensors are geometrical objects over vector spaces, whose coordinates obey certain laws of transformation under change of basis. Vectors are simple and well-known examples of tensors, but there is much more to tensor theory than vectors.

It stands to reason, then, that a tensor field is a set of tensors associated with every point in space: for instance, . It immediately follows that a scalar field is a zeroth-order tensor field, and a vector field is a first-order tensor field. Most tensor fields encountered in physics are smoothly varying and differentiable.

2020年9月7日 · 在数学，物理和工程上，张量场（tensor field）是一个的非常一般化的几何变量的概念。 它被用在 微分几何 和 流形 的理论中，在 代数几何 中，在 广义相对论 中，在材料的 应力 和 应变 的分析中，和在物理科学和工程的无数应用中。

In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was ...

What is the difference between a tensor and tensor field? Usually, a tensor field of a manifold M M is an assignment of a tensor to each point of M M. Just like a vector field of M M gives you a vector at a particular point of M M.

We call these product fields "tensors" of type (2, 0), (1, 1), and (0, 2) respectively. Definition 5.1 A tensor field of type (2, 0) on the n-dimensional smooth manifold M associates with each chart x a collection of n 2 smooth functions T ij (x 1, x 2, . . . , x n) which satisfy the …

In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors.

In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences.