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Tensor

{wiki}

A <multilinear form> with a <domain (function)> that looks like:
$$
V^m \times {V*}^n \to \R
$$
where $V*$ is the <dual space>.

Because a tensor is a <multilinear form>, it can be fully specified by how it act on all combinations of basis sets, which can be done in terms of components. We refer to each component as:
$$
T_{i_1 \ldots i_m}^{j_1 \ldots j_n} = T(e_{i_1}, \ldots, e_{i_m}, e^{j_1}, \ldots, e^{j_m})
$$
where we remember that the raised indices refer <dual vector>.

Some examples:
* <Levi-Civita symbol as a tensor>
* <a linear map is a (1,1) tensor>


Tensor

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