definition of interior multiplication of antisymmetric-tensor by vector
Topics
About: vectors space
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of interior multiplication of antisymmetric-tensor by vector.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\text{ the fields }\}\)
\( \{V, W\}\): \(\subseteq \{\text{ the } F \text{ vectors spaces }\}\)
\( \Lambda_k (V: W)\): \(= \text{ the } k \text{ -antisymmetric-tensors space }\)
\( \Lambda_{k - 1} (V: W)\): \(= \text{ the } k - 1 \text{ -antisymmetric-tensors space }\)
\( v\): \(\in V\)
\(*i_v\): \(: \Lambda_k (V: W) \to \Lambda_{k - 1} (V: W), t (\bullet) \mapsto t (v, \bullet)\), \(\in \{\text{ the } F \text{ linear maps }\}\)
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Conditions:
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2: Note
Let us see that \(i_v\) is indeed into \(\Lambda_{k - 1} (V: W)\).
\(i_v (t): V_2 \times ... \times V_k \to W\).
\(i_v (t) (v_2, ..., r v_j + r' v'_j, ..., v_k) = t (v, v_2, ..., r v_j + r' v'_j, ..., v_k) = r t (v, v_2, ..., v_j, ..., v_k) + r' t (v, v_2, ..., v'_j, ..., v_k) = r i_v (t) (v_2, ..., v_j, ..., v_k) + r' i_v (t) (v_2, ..., v'_j, ..., v_k)\).
So, \(i_v (t) \in L (V, ..., V: W)\).
For any permutation \(\sigma\) of \((2, ..., k)\), \(i_v (t) (v_{\sigma_2}, ..., v_{\sigma_k}) = t (v, v_{\sigma_2}, ..., v_{\sigma_k}) = sgn \sigma' t (v, v_2, ..., v_k)\) where \(\sigma'\) is the permutation of \((1, ..., k)\) that fixes \(1\) and permutates \((2, ..., k)\) by \(\sigma\), but \(sgn \sigma' = sgn \sigma\), so, \(= sgn \sigma t (v, v_2, ..., v_k) = sgn \sigma i_v (t) (v_2, ..., v_k)\), which means that \(i_v (t) \in \Lambda_{k - 1} (V: W)\).
Let us see that \(i_v\) is indeed an \(F\) linear map.
\((i_v (r t + r' t')) (v_2, ..., v_k) = (r t + r' t') (v, v_2, ..., v_k) = r t (v, v_2, ..., v_k) + r' t' (v, v_2, ..., v_k)\).
\((r i_v (t) + r' i_v (t')) (v_2, ..., v_k) = (r i_v (t)) (v_2, ..., v_k) + (r' i_v (t')) (v_2, ..., v_k) = r (i_v (t)) (v_2, ..., v_k) + r' (i_v (t')) (v_2, ..., v_k) = r t (v, v_2, ..., v_k) + r' t' (v, v_2, ..., v_k) \).
That means that \(i_v (r t + r' t') = r i_v (t) + r' i_v (t')\).
Usually, this concept is used for \(W = F\), but \(W \neq F\) is not invalid in any way.
