Tensor field
Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in material object, and in numerous applications in the physical sciences.
As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a magnitude and a direction, like velocity), a tensor field is a generalization of a scalar field and a vector field that assigns, respectively, a scalar or vector to each point of space.
Equivalently, it is a collection of elements Tx ∈ Vx⊗p ⊗ (Vx*)⊗q for all points x ∈ M, arranging into a smooth map T : M → V⊗p ⊗ (V*)⊗q.
Often we take V = TM to be the tangent bundle of M. Intuitively, a vector field is best visualized as an "arrow" attached to each point of a region, with variable length and direction.
One example of a vector field on a curved space is a weather map showing horizontal wind velocity at each point of the Earth's surface.
In general, we want to specify tensor fields in a coordinate-independent way: It should exist independently of latitude and longitude, or whatever particular "cartographic projection" we are using to introduce numerical coordinates.
Following Schouten (1951) and McConnell (1957), the concept of a tensor relies on a concept of a reference frame (or coordinate system), which may be fixed (relative to some background reference frame), but in general may be allowed to vary within some class of transformations of these coordinate systems.
may be subjected to arbitrary affine transformations: (with n-dimensional indices, summation implied).
More generally, the coordinates of a tensor of valence (p,q) have p upper indices and q lower indices, with the transformation law being The concept of a tensor field may be obtained by specializing the allowed coordinate transformations to be smooth (or differentiable, analytic, etc.).
Starting with the tangent bundle (the bundle of tangent spaces) the whole apparatus explained at component-free treatment of tensors carries over in a routine way – again independently of coordinates, as mentioned in the introduction.
(see the section on notation above) as a single space — a module over the ring of smooth functions, C∞(M), by pointwise scalar multiplication.
The notions of multilinearity and tensor products extend easily to the case of modules over any commutative ring.
-multilinear forms This kind of multilinearity implicitly expresses the fact that we're really dealing with a pointwise-defined object, i.e. a tensor field, as opposed to a function which, even when evaluated at a single point, depends on all the values of vector fields and 1-forms simultaneously.
A frequent example application of this general rule is showing that the Levi-Civita connection, which is a mapping of smooth vector fields
Nevertheless, it must be stressed that even though it is not a tensor field, it still qualifies as a geometric object with a component-free interpretation.
Differential forms, used in defining integration on manifolds, are a type of tensor field.
In theoretical physics and other fields, differential equations posed in terms of tensor fields provide a very general way to express relationships that are both geometric in nature (guaranteed by the tensor nature) and conventionally linked to differential calculus.
Even to formulate such equations requires a fresh notion, the covariant derivative.
(To be strictly accurate, one should also apply the absolute value to the transition functions – this makes little difference for an orientable manifold.)
One feature of the bundle of densities (again assuming orientability) L is that Ls is well-defined for real number values of s; this can be read from the transition functions, which take strictly positive real values.
In general we can take sections of W, the tensor product of V with Ls, and consider tensor density fields with weight s. Half-densities are applied in areas such as defining integral operators on manifolds, and geometric quantization.
It gives the consistency required to define the tangent bundle in an intrinsic way.
Implicit in defining tensors by how they transform under a coordinate change is the kind of self-consistency the cocycle expresses.
Geometers have not been in any doubt about the geometric nature of tensor quantities; this kind of descent argument justifies abstractly the whole theory.
The concept of a tensor field can be generalized by considering objects that transform differently.
An object that transforms as an ordinary tensor field under coordinate transformations, except that it is also multiplied by the determinant of the Jacobian of the inverse coordinate transformation to the wth power, is called a tensor density with weight w.[4] Invariantly, in the language of multilinear algebra, one can think of tensor densities as multilinear maps taking their values in a density bundle such as the (1-dimensional) space of n-forms (where n is the dimension of the space), as opposed to taking their values in just R. Higher "weights" then just correspond to taking additional tensor products with this space in the range.
Scalar 1-densities are especially important because it makes sense to define their integral over a manifold.
The most common example of a scalar 1-density is the volume element, which in the presence of a metric tensor g is the square root of its determinant in coordinates, denoted
While locally the more general transformation law can indeed be used to recognise these tensors, there is a global question that arises, reflecting that in the transformation law one may write either the Jacobian determinant, or its absolute value.
Restricting to changes of coordinates with positive Jacobian determinant is possible on orientable manifolds, because there is a consistent global way to eliminate the minus signs; but otherwise the line bundle of densities and the line bundle of n-forms are distinct.
