A Tensor

Tensor - Wikipedia

1. (anatomy) A muscle that stretches a part, or renders it tense.
2. (mathematics, linear algebra, physics) A mathematical object that describes linear relations on scalars, vectors, matrices and other tensors, and is represented as a multidimensional array.^[2]
3. (mathematics, obsolete) A norm operation on the quaternion algebra.

Usage notes

(mathematics, linear algebra):

• The array's dimensionality (number of indices needed to label a component) is called its order (also degree or rank).
• Tensors operate in the context of a vector space and thus within a choice of basis vectors, but, because they express relationships between vectors, must be independent of any given choice of basis. This independence takes the form of a law of covariant and/or contravariant transformation that relates the arrays computed in different bases. The precise form of the transformation law determines the type (or valence) of the tensor. The tensor type is a pair of natural numbers (n, m), where n is the number of contravariant indices and m the number of covariant indices. The total order of the tensor is the sum n + m.

Physics

A tensor is a mathematical object that describes the relationship between different geometric or physical quantities. It's a generalized container for data that ensures the underlying physical reality remains consistent under rotation coordinate system or changing the reference frame.

Tensor Rank

(or orders), it describes how many directions are needed to express the quantity.

• Rank 0 Tensor (Scalar): Requires zero directions. It is just a magnitude. Examples include mass, temperature, or volume.
• Rank 1 Tensor (Vector): Requires one direction. It has both magnitude and a single direction. Examples include velocity or a point force acting on a beam.
• Rank 2 Tensor: Requires two directions. This is where tensors become incredibly useful in statics and solid mechanics.

The Mechanical Stress Example

To understand a Rank 2 tensor, consider mechanical stress inside a structural beam. If you analyze a microscopic cubic element within that beam, a simple force vector isn't enough to describe the stress. You need to know two specific directions:

1. The direction of the force being applied.
2. The normal direction of the surface plane that the force is acting upon.

Because it requires two directions to fully describe, stress is a Rank 2 tensor, known as the Cauchy stress tensor. In 3D space, this tensor is represented by a \(3 \times 3\) matrix containing 9 components (3 normal stresses and 6 shear stresses):

\[\sigma = \begin{bmatrix} \sigma_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_{zz} \end{bmatrix}\]

Here, \(\sigma\) represents normal stresses (pushing or pulling directly against the face), and \(\tau\) represents shear stresses (sliding parallel to the face). The subscripts denote the two directions: the first letter is the face the force acts on, and the second is the direction of the force itself.

Higher-order tensors exist as well (for example, the elasticity tensor connecting stress and strain is a Rank 4 tensor, containing 81 components in 3D), but the core principle remains the same: tensors map vectors to other vectors to perfectly describe multi-dimensional physical states.