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Contraction (mathematics)
In mathematics, contraction has two meanings:
- Contraction of a tensor. It occurs when a pair of literal indices (one a subscript, the other a superscript) of a mixed tensor are set equal to each other so that a summation over that index takes place (due to the Einstein summation convention). The result is another tensor whose rank is reduced by 2.
- If a tensor is dyadic then its contraction is a scalar obtained by dotting each pair of base vectors in each dyad. E.g. Let <math>
\mathbf{T} = T^i{}_j \mathbf{e_i e^j} </math> be a dyadic tensor, then its contraction is <math> T^i {}_j \mathbf{e_i} \cdot \mathbf{e^j} = T^i {}_j \delta_i^j
T^j {}_j T^1 {}_1 + T^2 {}_2 + T^3 {}_3 </math>,
- a scalar of rank 0.
- E.g. Let <math> \mathbf{T} = \mathbf{e^i e^j} </math> be a dyadic tensor.
- This tensor does not contract; if its base vectors are dotted the result is the contravariant metric tensor, <math> g^{ij}= \mathbf{e^i} \cdot
\mathbf{e^j} </math>, whose rank is 2.
- References. Mathematical Physics by Donald H. Menzel. Dover Publications, New York.