Tensor

A $(𝑝, 𝑞)$ -tensor on a finite-dimensional vector space $𝑉$ is a multilinear object which “eats” $𝑝$ covectors and $𝑞$ vectors and spits out a scalar. multi Thus a $(𝑝, 𝑞)$ -tensor $𝑇$ on $𝑉$ is a multilinear map

𝑇 : 𝑉 * \times \dots \times 𝑉 * ⏟__⏟__⏟ 𝑝 \times 𝑉 \times \dots \times 𝑉 ⏟__⏟__⏟ 𝑞 \to 𝕂

or equivalently, and element of the tensor product space

𝑇 \in 𝑉 \otimes \dots \otimes 𝑉 ⏟__⏟__⏟ 𝑝 \otimes 𝑉 * \otimes \dots \otimes 𝑉 * ⏟__⏟__⏟ 𝑞 = T 𝑝 𝑞 𝑉

In infinite-dimensional contexts, it is necessary to use the latter definition.

Further terminology

Symmetric tensor
Antisymmetric tensor

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Tensor

Further terminology

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