173 k-Fold Product
A Cartesian product of k copies of one set that is used as the domain of a multilinear mapping of k inputs.
definition (k-Fold Product) A set constructed by the Cartesian product of \(k\) copies of a single set, \(X^k\), where the following condition applies:
The resulting set consists of all ordered \(k\)-tuples where
each coordinate is an element of the original set.
where
- \(k\) is a positive integer representing the number of repetitions.
- \(X\) is the underlying set.
- \(X^k\) denotes the \(k\)-fold Cartesian product \(X \times \dots \times X\).
Note:
- \(X\) may be a topological space.
- \(X\) may be a vector space.
- \(X^k\) is also written \(V^k\).
173.1 Elementary Example
173.1.1 Simple
The \(2\)-fold product is the Cartesian product of two copies of one set.
\[ V = \{ a,\ b,\ c \} \]
\[ V^{2} = V \times V \]
\[ (a,b),\ (b,a),\ (a,c) \in V^{2} \]
where
- \(V^{2}\) is the \(2\)-fold product.
- \(k = 2\) is the number of factors.
173.1.2 General
The \(3\)-fold product \(V^{3}\) is the set of ordered triples from \(V\).
\[ V = \{ a,\ b,\ c \} \]
\[ V^{3} = V \times V \times V \]
\[ (a,b,c),\ (c,b,a),\ (a,a,b) \in V^{3} \]
where
- \(V^{3}\) is the domain of a \(3\)-linear map on \(V\).
- Alternating Function
- Alternating k-linear function
- Boundary Operation
- Bundle
- Cartesian Product
- Constant Metric Tensor
- Contravariant Rank
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- Covariant, Contravariant, and Coordinate Systems
- Differential Forms
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- Edge of a Domain
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- Inverse Metric
- k-Covector
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- k-Fold Product
- k-form
- k-Tensor
- Lie
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