Alternating unit tensor: Difference between revisions

Revision as of 16:25, 7 November 2023

A mathematical function with symbol ε_ijk defined to switch between the discrete values of +1, 0, and -1, depending on the values of the three indices i, j, and k:

It is one of the tools used in Einstein's summation notation to handle operations equivalent to cross products in vector notation. For example: A × B = ε_ijkA_iB_jδ_k, where A_i and B_j represent vectors in summation notation, and δ_k is the unit vector. Subscripts or indices i, j, and k must each take on the values of 1, 2, and 3, representing the Cartesian directions of x, y, and z. The alternating unit tensor is useful to describe the Coriolis term in the equations of motion: ∂U_i/ ∂t = . . . + fε_ij3U_j + . . ., where f is the Coriolis parameter, U_i and U_j are wind components, and t is time. Summation notation is commonly used in atmospheric turbulence studies.
Stull, R. B. 1988. An Introduction to Boundary Layer Meteorology. 666 pp.

INDEX
A
B
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G
H
I
J
K
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M
N
O
P
Q
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S
T
U
V
W
X
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Z
0
1
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+{{Term
+|Display title=alternating unit tensor
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-<div class="termentry">
+A mathematical function with symbol &#x003b5;<sub>''ijk''</sub> defined to switch between the  discrete values of +1, 0, and -1, depending on the values of the three indices ''i'', ''j'', and ''k'':<br/> <blockquote>[[File:ams2001glos-Ae16.gif|link=|center|ams2001glos-Ae16]]</blockquote> It is one of the tools used in [[Einstein's summation notation]] to handle operations equivalent to  cross products in [[vector]] notation. For example: '''A''' &times; '''B''' = &#x003b5;<sub>''ijk''</sub>''A''<sub>''i''</sub>''B''<sub>''j''</sub>&#x003b4;<sub>''k''</sub>, where ''A''<sub>''i''</sub> and ''B''<sub>''j''</sub> represent  vectors in [[summation notation]], and &#x003b4;<sub>''k''</sub> is the unit vector. Subscripts or indices ''i'', ''j'', and ''k'' must  each take on the values of 1, 2, and 3, representing the Cartesian directions of ''x'', ''y'', and ''z''. The  alternating unit tensor is useful to describe the Coriolis term in the [[equations of motion]]: &part;''U''<sub>''i''</sub>/  &part;''t'' = . . . + ''f''&#x003b5;<sub>''ij''3</sub>''U''<sub>''j''</sub> + . . ., where ''f'' is the [[Coriolis parameter]], ''U''<sub>''i''</sub> and ''U''<sub>''j''</sub> are [[wind]] components,  and ''t'' is time. Summation notation is commonly used in [[atmospheric turbulence]] studies.<br/> Stull, R. B. 1988. An Introduction to Boundary Layer Meteorology.  666 pp.
-  <div class="term">
+}}
-== alternating unit tensor ==
+}}
-  </div>
-<div class="definition"><div class="short_definition">A mathematical function with symbol &#x003b5;<sub>''ijk''</sub> defined to switch between the  discrete values of +1, 0, and -1, depending on the values of the three indices ''i'', ''j'', and ''k'':</div><br/> <div class="paragraph"><div class="display-formula"><blockquote>[[File:ams2001glos-Ae16.gif|link=|center|ams2001glos-Ae16]]</blockquote></div> It is one of the tools used in [[Einstein's summation notation]] to handle operations equivalent to  cross products in [[vector]] notation. For example: '''A''' &times; '''B''' = &#x003b5;<sub>''ijk''</sub>''A''<sub>''i''</sub>''B''<sub>''j''</sub>&#x003b4;<sub>''k''</sub>, where ''A''<sub>''i''</sub> and ''B''<sub>''j''</sub> represent  vectors in [[summation notation]], and &#x003b4;<sub>''k''</sub> is the unit vector. Subscripts or indices ''i'', ''j'', and ''k'' must  each take on the values of 1, 2, and 3, representing the Cartesian directions of ''x'', ''y'', and ''z''. The  alternating unit tensor is useful to describe the Coriolis term in the [[equations of motion]]: &part;''U''<sub>''i''</sub>/  &part;''t'' = . . . + ''f''&#x003b5;<sub>''ij''3</sub>''U''<sub>''j''</sub> + . . ., where ''f'' is the [[Coriolis parameter]], ''U''<sub>''i''</sub> and ''U''<sub>''j''</sub> are [[wind]] components,  and ''t'' is time. Summation notation is commonly used in [[atmospheric turbulence]] studies.</div><br/> </div><div class="reference">Stull, R. B. 1988. An Introduction to Boundary Layer Meteorology.  666 pp. </div><br/>
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