Classical Treatment Of Tensors

The Einstein Notation is used throughout this page.
For help with notation, refer to the Table Of Mathematical Symbols .

A tensor is a generalization of the concepts of vectors and matrices. Tensors allow one to express physical laws in a form that applies to any coordinate system. For this reason, they are used extensively in Continuum Mechanics and the Theory Of Relativity .

A tensor is an Invariant multi-dimensional transformation, that takes forms in one coordinate system into another. It takes the form:

:

T^{\left[i_1,i_2,i_3,...i_n
ight]}_{\left[j_1,j_2,j_3,...j_m
ight]}

The new coordinate system is represented by being 'barred'(

\bar{x}^i

), and the old coordinate system is unbarred(

x^i

).

The upper indices are the Contravariant components, and the lower indices [

j_1,j_2,j_3,...j_n

are the Covariant components.

CONTRAVARIANT AND COVARIANT TENSORS

A contravariant tensor of order 1(

T^i

) is defined as:

:

\bar{T}^i = T^rrac{\partial \bar{x}^i}{\partial x^r}.

A covariant tensor of order 1(

T_i

) is defined as:

:

\bar{T}_i = T_rrac{\partial x^r}{\partial \bar{x}^i}.

GENERAL TENSORS

A multi-order (general) tensor is simply the Tensor Product of single order tensors:

:

T^{\left[i_1,i_2,...i_p
ight]}_{\left[j_1,j_2,...j_q
ight]} = T^{i_1} \otimes T^{i_2} ... \otimes T^{i_p} \otimes T_{j_1} \otimes T_{j_2} ... \otimes T_{j_q}

such that:

:

\bar{T}^{\left[i_1,i_2,...i_p
ight]}_{\left[j_1,j_2,...j_q
ight]} = 
T^{\left[r_1,r_2,...r_p
ight]}_{\left[s_1,s_2,...s_q
ight]}
rac{\partial \bar{x}^{i_1}}{\partial x^{r_1}}
rac{\partial \bar{x}^{i_2}}{\partial x^{r_2}}
...
rac{\partial \bar{x}^{i_p}}{\partial x^{r_p}}
rac{\partial x^{s_1}}{\partial \bar{x}^{j_1}}
rac{\partial x^{s_2}}{\partial \bar{x}^{j_2}}
...
rac{\partial x^{s_q}}{\partial \bar{x}^{j_q}}.

This is sometimes termed the tensor transformation law.

SEE ALSO

Tensor Product

Tensor Derivative

Absolute Differentiation

Curvature

Riemannian Geometry

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Classical Treatment Of Tensors