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diff --git a/wasm/examples/mediawiki-to-docx-with-equations/stdin b/wasm/examples/mediawiki-to-docx-with-equations/stdin new file mode 100644 index 000000000..18cb37bbb --- /dev/null +++ b/wasm/examples/mediawiki-to-docx-with-equations/stdin @@ -0,0 +1,8 @@ +Just as the components of a vector change when we change the [[basis (linear algebra)|basis]] of the vector space, the components of a tensor also change under such a transformation. Each type of tensor comes equipped with a ''transformation law'' that details how the components of the tensor respond to a [[change of basis]]. The components of a vector can respond in two distinct ways to a [[change of basis]] (see [[covariance and contravariance of vectors]]), where the new [[basis vectors]] <math>\mathbf{\hat{e}}_i</math> are expressed in terms of the old basis vectors <math>\mathbf{e}_j</math> as, +:<math>\mathbf{\hat{e}}_i = \sum_{j=1}^n \mathbf{e}_j R^j_i = \mathbf{e}_j R^j_i .</math> + +Here ''R''<sup>'' j''</sup><sub>''i''</sub> are the entries of the change of basis matrix, and in the rightmost expression the [[summation]] sign was suppressed: this is the [[Einstein summation convention]], which will be used throughout this article.<ref group="Note">The Einstein summation convention, in brief, requires the sum to be taken over all values of the index whenever the same symbol appears as a subscript and superscript in the same term. For example, under this convention <math>B_i C^i = B_1 C^1 + B_2 C^2 + \cdots B_n C^n</math></ref> The components ''v''<sup>''i''</sup> of a column vector '''v''' transform with the [[matrix inverse|inverse]] of the matrix ''R'', +:<math>\hat{v}^i = \left(R^{-1}\right)^i_j v^j,</math> + +where the hat denotes the components in the new basis. This is called a ''contravariant'' transformation law, because the vector components transform by the ''inverse'' of the change of basis. In contrast, the components, ''w''<sub>''i''</sub>, of a covector (or row vector), '''w''', transform with the matrix ''R'' itself, +:<math>\hat{w}_i = w_j R^j_i .</math>
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