Tensors and Tensorial Algebra

Basic concepts ans principles

When we handle multiple variable in Linear algebras we need to use a generalized concept of Vector, matrix and constants, this concep is called Tensor.
To define it we must begin to remaind that all finite-dimension vector space V is isomorphic to Rn, so from now we can consider in fact V ≡Rn.

Anyone linear application f: Rn → R is in the form

f(x)=vtx, con v, x∈Rn

Ie, Dual space of Rn is really itself (or they are isomorphics by transpose vt to v ).

Lets B={e1, ...,en} a Basis of V, then B*={e1, ...,en} is a basis of Dual V*, characterized by

ejei = 0, j≠i
ejei = 1, j=i

This isomorphism, we call φ between V and its dual V*, allows us to consider any linear application in the form.

f: V x ...(n times)... x V → V

as

F: V* x V x ...(n times)... x V → R

Being F=φ◊f (composition of f and φ) , ie, after applying f, we apply the dual element to obtain a scalar.

So, we can consider Scalar Multilineal Functions simply, by multiplying by elements of the dual

Tensor definition


It is called s-contravariant and n-covariant Tensor or s-times contravariant n-times covariant tensor, or tensor kind (s, n) to a multilinear Map like

T: V* x ...(s times)... x V* x V x ...(n times)... x V → R

If a tensor is (0, n) type, we say simply that it is n times covariant

If a tensor is type (s, 0) say simply that it is s times contravariant
So, the endomorphism that we considered at diagonalizations or Jordan Cannonical form sections were
F : V → V
x∈V → F(x) = Ax∈V
by our definition F is an (1,1) tensor.

Also a vector is a tensor kind (1,0) because it applies any row vector (dual element) to a constant.

The scalar product in V is a tensor of type (0,2), because it applies two vectors to a constant.

Finally we consider that a constant is a (0,0) tensor.

Extended Theory

Let T be a (r, s) tensor, therefore we will consider ∏ = V* x ...(r times)... x V* x V x ...(s times)... x V
T: ∏ → R
u1, ...,ur∈V*, v1, ...,vs ∈V → F(u1, ...,ur,v1, ...,vs)∈ R

Then, if B={e1, ...,en} is an Basis of V and B*={e1,...,en} is a basis of dual V*, we can consider the basis of ∏ as β=B*∪B, then we define the coordinates of T at basis β as

= T(ej1,...,ejr,ei1,...,e1r)

Therefore, a (r,s) tensor on Rn has nr+s components

Given two bases in ∏, β y β there is a 'change of basis matrix' for covariant and contravariant coordinates given respectively by , Then given a tensor T to change the Coordinate of β a β'



Note the Einstein summation convention

There is a tensorial product denoted por ⊗, defined as follows, if T is a tensor (r, s) and S is a tensor (m, n), tensorial product is a (r + m, s + n) tensor consisting of

T⊗S(u1,...,ur, u1, ...,um, v1, ...,vs,v1,...,vn) = T(u1,...,ur,v1,...,vs) .S(u1, ...,um,v1,...,vn)

The notation tensotrial can get very complicated because we are dealing with covariant and contravariant coordinates of different dimensions, such as the Riemann tensor is a tensor (1.3) as it applies a contravector of R4 (z1,z2,z3,z4) and 3 vectors (u1,u2,u3,u4) (v1,v2,v3,v4) (w1,w2,w3,w4) to denote the application of the tensor in the vectors would have to write

    ∑4i=14j=14k=14l=1Rlijk zluivjwk

withEinstein summation convention this equation is reduced to

    Rlijk zluivjwk

ie, Einstein summation convention consists in a duplicate index and dubindex indicates addition, for example a tensor given by vijwkl is a (2,2) tensor, showever vikwkl is a (1,1) tensor.

Therefore we can consider a constant k is a (0,0) tensor, because it is the contraction of a (1,0) tensor with a (0,1) tensor

k=viwi

Tensors in Manifolds

In a differentiable manifold M, one can define a tensor field or just a tensor of kind (r,s) as an application which assigns at each point p of a manifold an (r,s) tensor such us

T: Tp(M)* x ...(r veces)... x Tp(M)* x Tp(M) x ...(s veces)... x Tp(M) → R

That is, taking the tangent vector space as a vector space where tensor operates.

Coordinate tensor transformations

Given two charts C = (φ=(x1,...,xm), U) K = (ψ=(y1,...,ym), V)
There exists an application ψoφ-1 wich transforms coodinates (x1,...,xm) into coordinates (y1,...,ym) and its differential matrix is written as



and its inverse



Each chart have the vector fields

and

and

Respectively and the relationship between them is

1)

2)


A very important tensor in differential geometry is called Metric Tensor. We denote it by G=gij

Given an element v=vi of V, then v is a (1,0)tensor. It is possible to apply the Metric Tensor as follows

wj=gijvi

Obtaining a (0,1) tensor w, ie, an dual space element. We will call vi the contravariant coordiantes of v and wj will call covariants ones.

Finally, this process of matching indexes with superindexes and sum them (again the Einstein summation convention) is called contraction.

Riemannian Manifolds

Is called a Riemannian a manifold to a differentiable manifold join a metric tensor acting on it, whose coefficients matrix is definite-positive,

When matrix coefficient of metric tensor is positive-semidefinite, is said that manifold is Semirimannian






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