Tensor Field · aiwiki.page

A tensor field assigns a tensor of a specified type to every point of a manifold or a region within it. In differential geometry, tensor fields are normally assumed to be smooth: their components vary smoothly in every smooth coordinate system. They generalize scalar and vector fields and provide a coordinate-independent language for describing geometry and physical quantities. The field itself is a geometric object; its component arrays depend on the coordinates chosen to represent it. (damtp.cam.ac.uk)

Mathematical definition

Let $M$ be a smooth manifold of dimension $n$ . At each point $p$ , the tangent space $T_pM$ is a vector space whose dual space $T_p^*M$ consists of covectors. A tensor field of type $(r,s)$ assigns an element

T_p\in (T_pM)^{\otimes r}\otimes(T_p^*M)^{\otimes s}

to every $p\in M$ . Here $\otimes$ denotes the tensor product, and a zeroth tensor power is interpreted as the real numbers. Equivalently, $T_p$ is a multilinear map taking $r$ covectors and $s$ tangent vectors to a scalar. (math.toronto.edu)

The integers $r$ and $s$ count the contravariant and covariant indices, respectively. Their sum is the tensor's order, sometimes called its rank; this usage differs from matrix rank. Collecting these tensor spaces over all points produces a tensor bundle. A tensor field is a section of this bundle: it selects one element from the fiber above each point. This formulation matters because tensors at different points belong to different vector spaces, rather than necessarily to one fixed space. (math.toronto.edu)

Components and coordinate changes

In local coordinates $x^1,\ldots,x^n$ , the coordinate tangent vectors $\partial_i=\partial/\partial x^i$ and their dual covectors $dx^i$ supply a basis. A tensor field is written

T= T^{i_1\cdots i_r}{}_{j_1\cdots j_s}(x)\, \partial_{i_1}\otimes\cdots\otimes\partial_{i_r} \otimes dx^{j_1}\otimes\cdots\otimes dx^{j_s}.

Repeated indices are summed according to the Einstein summation convention. Smoothness means that these component functions are smooth. An unrestricted tensor has $n^{r+s}$ components at each point, although symmetry conditions can reduce the number of independent components. (damtp.cam.ac.uk)

Under a coordinate change from $x$ to $x'$ , each upper index transforms with a factor $\partial x'^a/\partial x^i$ , and each lower index with an inverse factor $\partial x^j/\partial x'^b$ . These factors are entries of the Jacobian matrix and its inverse. For a type- $(1,1)$ field,

T'^a{}_b= \frac{\partial x'^a}{\partial x^i} \frac{\partial x^j}{\partial x'^b} T^i{}_j.

The transformation compensates for the changing basis, leaving the underlying tensor unchanged. Consequently, an arbitrary collection of component functions in overlapping charts defines a tensor field only if those functions satisfy the appropriate transformation law. (damtp.cam.ac.uk)

Basic examples

A type- $(0,0)$ tensor field is a scalar field, represented by a function on $M$ . A vector field has type $(1,0)$ , while a covector field, or one-form, has type $(0,1)$ . For a smooth scalar function $f$ , its differential

df=(\partial_i f)\,dx^i

is a covector field. Turning this differential into a gradient vector requires a metric to identify covectors with vectors. (damtp.cam.ac.uk)

A metric tensor is a symmetric, nondegenerate type- $(0,2)$ field. A Riemannian metric is positive definite, whereas the metric of relativistic spacetime has Lorentzian signature. Metrics define lengths and other geometric measurements and allow indices to be raised or lowered. For example, $V_i=g_{ij}V^j$ . Coordinate-dependent metric components do not by themselves imply curvature: flat Euclidean space also has varying metric components in spherical coordinates. (preposterousuniverse.com)

A differential $k$ -form is a completely antisymmetric covariant tensor field of order $k$ . Forms support coordinate-independent differentiation and integration, making them especially useful in geometry and field theory. (preposterousuniverse.com)

Algebra and differentiation

Tensor operations extend to fields point by point. Fields of the same type can be added and multiplied by smooth scalar functions. Tensor products combine types: multiplying fields of types $(r,s)$ and $(u,v)$ produces type $(r+u,s+v)$ . Contraction pairs an upper and a lower index, reducing both counts by one. Thus $T^i{}_i$ is a scalar field when $T$ has type $(1,1)$ . (damtp.cam.ac.uk)

Ordinary partial derivatives of tensor components generally do not form a tensor under nonlinear coordinate changes. A covariant derivative corrects this using connection coefficients. For a vector field,

\nabla_i V^j=\partial_i V^j+\Gamma^j{}_{ik}V^k.

The resulting $\nabla V$ has type $(1,1)$ ; more generally, covariant differentiation sends type $(r,s)$ to type $(r,s+1)$ . The coefficients $\Gamma^j{}_{ik}$ themselves are not tensor components. A metric determines a unique torsion-free, metric-compatible Levi-Civita connection. Its curvature is expressed by the Riemann curvature tensor. (preposterousuniverse.com)

Differential forms have another natural operation, the exterior derivative, which needs neither a metric nor a connection. It maps $k$ -forms to $(k+1)$ -forms and satisfies $d^2=0$ . (preposterousuniverse.com)

Physical applications

In electromagnetism, the electromagnetic field strength is an antisymmetric tensor field combining electric and magnetic components. This permits Maxwell's equations to be expressed in covariant form, with transformation behavior made explicit. (preposterousuniverse.com)

In general relativity, the spacetime metric is a dynamical tensor field. The Einstein field equations relate geometric tensors constructed from the metric to the stress–energy tensor, which describes energy, momentum, and stress. Written as tensor equations, these relations retain their meaning under changes of coordinates rather than depending on a particular coordinate representation. (damtp.cam.ac.uk)

References

Geometry and Topology Imath.toronto.edu
Tensor Bundles on Manifoldsmath.utoronto.ca
Lecture Notes on General Relativity: Manifoldspreposterousuniverse.com
Lecture Notes on General Relativity: Special Relativity and Flat Spacetimepreposterousuniverse.com
Lecture Notes on General Relativity: Curvaturepreposterousuniverse.com