View source for mai'u'ei

==== valsi ====
mai'u'ei
==== type ====
experimental cmavo
==== creator ====
[[personal/krtisfranks|krtisfranks]]
==== time entered ====
Tue Oct 29 01:41:59 2024
 
== English ==
=== Definition #73677 - Preferred ===
 
==== selma'o ====
VUhU3
==== definition ====

mekso unary operator: Levi-Civita symbol; for input <span class="MATH"><i>n</i></span>-tuple  MATH
 $(a_1, a_2,$
 
<span class="MATH">(<i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>,</span> ... <span class="MATH">, <i>a</i><sub>n</sub>)</span>, where <span class="MATH"><i>n</i></span> is a strictly positive integer, the output is  MATH
 $\varepsilon_{a_1, a_2, ..., a_n}$
 
<span class="MATH"><img alt="$\varepsilon_{{a_1, a_2, ..., a_n}}^{}$" src="/jbovlaste_export/vDiUrtORQI/img1.png" style="height: 1.96ex; vertical-align: -1.01ex; "/></span>, where  MATH
 $\varepsilon$
 
<span class="MATH"><img alt="$\varepsilon$" src="/jbovlaste_export/vDiUrtORQI/img2.png" style="height: 1.12ex; vertical-align: -0.12ex; "/></span> is the Levi-Civita symbol under the convention of mapping <span class="MATH">(1, 2,</span> ...<span class="MATH">, <i>n</i>)</span> to <span class="MATH">1</span>.

==== notes ====

If any at least two elements of the input tuple are mutually equal (id est: if there exists <span class="MATH"><i>i</i></span> and <span class="MATH"><i>j</i></span> which are strictly positive integers which do not exceed <span class="MATH"><i>n</i></span> such that <span class="MATH"><i>a</i><sub>i</sub> = <i>a</i><sub>j</sub></span>, where the input tuple is as defined supra), then the output is <span class="MATH">0</span>; else, the output is sgn MATH
 $((a_1, a_2,$
 
<span class="MATH">((<i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>,</span> ... <span class="MATH">, <i>a</i><sub>n</sub>))</span>, where the input to the sgn function there is a permutation denoted in one-line permutation notation. This format is somewhat consistent with index notation in tensor calculus, but treats the indices as ordered entries of a tuple. More generally, the Levi-Civita symbol may be defined for any function <span class="MATH"><i>f</i></span> which maps space <span class="MATH"><i>S</i></span> to <span class="MATH"><i>S</i></span>; in such case, the input could be <span class="MATH"><i>f</i></span> instead of the aforementioned tuple. Covariant and contravariant Levi-Civita tensors can also be defined. See also: "[[mai'u'e|mai'u'e]]", "[[mai'u|mai'u]]".

==== gloss words ====

<ul><li>[[natlang/en/Levi-Civita symbol|Levi-Civita symbol]]</li></ul>

==== created by ====
[[personal/krtisfranks|krtisfranks]]
==== vote information ====
1
     
==== time ====
Tue Oct 29 01:47:41 2024



<br/><font size="+1">Examples</font>
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=== Etymology ===