Tensor field

In mathematics and physics, a tensor field is a special kind of function that gives a tensor to every point in a certain region of space. This space can be a mathematical space like a Euclidean space or a Riemannian manifold, or it can be the real physical space we live in. Tensor fields are very important in many areas of science and math.

Tensor fields are used in differential geometry, algebraic geometry, and general relativity. They help scientists understand things like stress and strain in materials. Since a tensor can describe more complex information than a simple number or a direction, a tensor field does the same for every point in space.

For example, the Riemann curvature tensor is a famous tensor field. It shows how space is curved at each point, which is important in Einstein’s theory of gravity. Just like a scalar field gives a number to each point and a vector field gives a direction to each point, a tensor field gives more detailed information. This makes tensor fields useful tools in science.

Definition

A tensor field is a way to give a tensor to every point in space. Imagine giving each point a small set of numbers that tell us something about that point.

Tensor fields are important in physics and math. They help us understand how materials stretch or bend, and they are used in ideas about space and time. They build on simpler ideas like scalars (single numbers) and vectors (numbers with direction), making things more detailed and useful.

Geometric introduction

Imagine a vector field like a weather map that shows wind speed and direction at every point on Earth's surface — each point has an "arrow" representing the wind there.

Tensor fields are more complex. They assign a set of numbers or relationships to each point in space. These relationships can describe how distances change on curved surfaces, without needing to use specific map coordinates.

Via coordinate transitions

The idea of a tensor depends on having a reference frame, or a coordinate system. This system can change in certain ways. For example, in a space with n dimensions, the coordinates can be shifted or stretched using special rules.

When we change these coordinates, different types of mathematical objects, like vectors, change in specific ways to keep certain quantities unchanged. This helps ensure that important calculations stay the same, no matter how we describe the space.

Tensor bundles

A tensor bundle is a special structure called a fiber bundle. In a tensor bundle, each small piece, or fiber, is made from copies of spaces called tangent space and cotangent space. These fibers are vector spaces, so a tensor bundle is a type of vector bundle.

Tensor fields are special parts of these tensor bundles. They give a tensor — a math object that generalizes numbers and vectors — to each point in a space. This way, the math stays coordinate-free and focuses on the space's geometry.

Notation

The notation for tensor fields can look similar to the notation for tensor spaces, which can be confusing. For example, the tangent bundle TM might be written in a special way to show that it relates to vector fields on a manifold M.

Sometimes, curly letters are used to represent the set of smoothly changing tensor fields on M. These tensor fields are parts of a tensor bundle and change smoothly from point to point.

Tensor fields as multilinear forms

Tensor fields can be thought of in a more abstract way. They are special kinds of maps. These maps take vector fields and covector fields as inputs and give out smooth functions as outputs. This means that tensor fields behave in an organized and predictable way at each point in space.

For example, a covector field, which is a type of tensor field, can act on a vector field to produce a smooth function. This action follows specific rules that make it linear and consistent across the whole space. These ideas help mathematicians and physicists describe complex relationships in geometry and physics using a clear and structured approach.

Applications

Tensor fields are useful in science and math. In physics, the stress–energy tensor shows how matter and energy shape space, as explained by Einstein's theory of general relativity. In electromagnetism, electric and magnetic fields are brought together into an electromagnetic tensor field. Differential forms, which help us understand integration on complex shapes, are a type of tensor field.

Tensor calculus

In theoretical physics and other areas, differential equations that use tensor fields help describe important relationships. To create these equations, a special idea called the covariant derivative is used. This idea helps explain how a tensor field changes along a vector field. The original idea was called absolute differential calculus. It later became known as tensor calculus, and it helped discover the geometric idea of a connection.

Twisting by a line bundle

A tensor field can be extended by adding something called a line bundle. This helps create a special kind of tensor field called a tensor density. Tensor densities are useful in mathematics, especially when working with certain shapes.

One interesting use of tensor densities is in half-densities, where we can use values like one-half. These are important in advanced math topics like integral operators and geometric quantization.

Flat case

When we think about a special space called Euclidean space, a tensor field works like a tensor at one point. This idea is used in real-life problems, but it has some limits with certain tensors called tensor densities.

Cocycles and chain rules

The chain rule in multivariable calculus helps us understand how tensors change when we use different coordinates. It makes sure tensors work the same in all coordinate systems. This is important for tensor fields.

We can think of the chain rule as a special math object called a 1-cocycle. This helps mathematicians define structures like the tangent bundle in a natural way. Other tensor structures also follow similar rules. This makes them meaningful in geometry.

Generalizations

Tensor densities

Main article: Tensor density

A tensor field can be made more flexible by looking at special objects called tensor densities. These work like regular tensors but also change in a special way depending on how we look at space.

One important example is the volume element. This helps us measure space in physics, and it is especially useful in theories like general relativity. It changes in a predictable way when we change our view of space, which helps make complex calculations easier.