Tensor field

Tensor field is a mathematical construct that generalizes scalar fields and vector fields to higher dimensions. In the context of differential geometry and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a manifold). Tensor fields are crucial in the formulation of theories in physics, including general relativity and field theory, as they can represent concepts such as stress, strain, and curvature in a way that is independent of coordinate choice.

Definition

A tensor field of type (r, s) over a manifold M is a function that assigns to each point p in M an (r, s)-tensor on the tangent space at p, in a smooth manner. Here, r denotes the number of contravariant indices and s the number of covariant indices of the tensor. The collection of all such tensor fields on M forms a vector space, often denoted as \(\mathcal{T}^r_s(M)\).

Examples

• The simplest example of a tensor field is a Scalar field, which assigns a scalar (a tensor of type (0, 0)) to every point in a space. Temperature distribution in a room can be modeled as a scalar field.
• A Vector field is a tensor field of type (1, 0), assigning a vector to each point in space. Examples include the velocity field of a flowing fluid and the electric field around charged objects.
• The Metric tensor is a fundamental tensor field of type (0, 2) in differential geometry and general relativity, providing a way to measure distances and angles on manifolds.
• The Stress tensor in continuum mechanics is an example of a tensor field of type (0, 2), representing the internal forces that neighboring particles of a continuous material exert on each other.

Applications

Tensor fields are indispensable in many areas of physics and engineering:

• In General Relativity, the Einstein field equations describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. Here, the metric tensor is a key component.
• In Fluid dynamics, tensor fields describe the stress and strain within fluid flows.
• In Elasticity, the stress and strain tensor fields model how materials deform under external forces.

Mathematical Properties

Tensor fields possess several important mathematical properties:

• They can be added together and multiplied by scalars, making the set of all tensor fields of a given type over a manifold a vector space.
• The Covariant derivative allows for the differentiation of tensor fields with respect to vector fields, providing a way to express the rate of change of tensor fields along curves in the manifold.
• The Lie derivative measures the change of a tensor field along the flow of another vector field, playing a crucial role in dynamics and symmetry analysis.

See Also

• Differential geometry
• Tensor calculus
• Manifold
• Riemannian geometry