Interior product: Difference between revisions
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== Definition == | == Definition == | ||
The '''interior product''' is a notion dual to the wedge product, so we will denote it using ∨ rather than ∧. To define it, we first discuss the multilinear map, or [[ | The '''interior product''' is a notion dual to the wedge product, so we will denote it using ∨ rather than ∧. To define it, we first discuss the multilinear map, or [[Wedgies and multivals|''n''-map]], a multival of rank ''n'' induces on a list of ''n'' monzos. | ||
Let ''W'' be a multival of rank ''n'', and ''m''<sub>1</sub>, ''m''<sub>2</sub>, ..., ''m''<sub>''n''</sub> be a group of ''n'' monzos. Take the wedge product of these monzos in exactly the same way as the wedge product of ''n'' vals, producing the multimonzo ''M''. Treating both ''M'' and ''W'' as ordinary vectors, take the dot product. This is the value of ''W''(''m''<sub>1</sub>, ''m''<sub>2</sub>, ..., ''m''<sub>''n''</sub>). | Let ''W'' be a multival of rank ''n'', and ''m''<sub>1</sub>, ''m''<sub>2</sub>, ..., ''m''<sub>''n''</sub> be a group of ''n'' monzos. Take the wedge product of these monzos in exactly the same way as the wedge product of ''n'' vals, producing the multimonzo ''M''. Treating both ''M'' and ''W'' as ordinary vectors, take the dot product. This is the value of ''W''(''m''<sub>1</sub>, ''m''<sub>2</sub>, ..., ''m''<sub>''n''</sub>). | ||