http://debianws.lexgopc.com/wiki143/index.php?action=history&feed=atom&title=Cartesian_tensor
Cartesian tensor - Revision history
2026-10-26T15:03:31Z
Revision history for this page on the wiki
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http://debianws.lexgopc.com/wiki143/index.php?title=Cartesian_tensor&diff=2057304&oldid=prev
imported>Localwerewolfenjoyer: /* growthexperiments-addlink-summary-summary:3|0|0 */
2025-06-28T15:44:55Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Previous revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:44, 28 June 2025</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l278">Line 278:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for all {{mvar|i}}, {{mvar|j}}.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>for all {{mvar|i}}, {{mvar|j}}.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>A vector is invariant under any change of basis, so if coordinates transform according to a [[transformation matrix]] {{math|'''L'''}}, the bases transform according to the [[matrix inverse]] {{math|'''L'''<sup>−1</sup>}}, and conversely if the coordinates transform according to inverse {{math|'''L'''<sup>−1</sup>}}, the bases transform according to the matrix {{math|'''L'''}}. The difference between each of these transformations is shown conventionally through the indices as superscripts for contravariance and subscripts for covariance, and the coordinates and bases are [[linear transformation|linearly transformed]] according to the following rules:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>A vector is invariant under any <ins style="font-weight: bold; text-decoration: none;">[[</ins>change of basis<ins style="font-weight: bold; text-decoration: none;">]]</ins>, so if coordinates transform according to a [[transformation matrix]] {{math|'''L'''}}, the bases transform according to the [[matrix inverse]] {{math|'''L'''<sup>−1</sup>}}, and conversely if the coordinates transform according to inverse {{math|'''L'''<sup>−1</sup>}}, the bases transform according to the matrix {{math|'''L'''}}. The difference between each of these transformations is shown conventionally through the indices as superscripts for contravariance and subscripts for covariance, and the coordinates and bases are [[linear transformation|linearly transformed]] according to the following rules:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| class="wikitable"</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| class="wikitable"</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l736">Line 736:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{pmatrix}</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{pmatrix}</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>so the pseudovector {{math|'''a'''}} can be written as an antisymmetric tensor. This transforms as a tensor, not a pseudotensor. For the mechanical example above for the tangential velocity of a rigid body, given by {{math|1='''v''' = '''''ω''''' × '''x'''}}, this can be rewritten as {{math|1='''v''' = '''Ω''' ⋅ '''x'''}} where {{math|'''Ω'''}} is the tensor corresponding to the pseudovector {{math|'''ω'''}}:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>so the pseudovector {{math|'''a'''}} can be written as an <ins style="font-weight: bold; text-decoration: none;">[[</ins>antisymmetric tensor<ins style="font-weight: bold; text-decoration: none;">]]</ins>. This transforms as a tensor, not a pseudotensor. For the mechanical example above for the tangential velocity of a rigid body, given by {{math|1='''v''' = '''''ω''''' × '''x'''}}, this can be rewritten as {{math|1='''v''' = '''Ω''' ⋅ '''x'''}} where {{math|'''Ω'''}} is the tensor corresponding to the pseudovector {{math|'''ω'''}}:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block">\boldsymbol{\Omega} = \begin{pmatrix}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block">\boldsymbol{\Omega} = \begin{pmatrix}</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block">\mathbf{A}\cdot\nabla = A_i \nabla_i </math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block">\mathbf{A}\cdot\nabla = A_i \nabla_i </math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>which could act on scalar or vector fields. In fact, if '''A''' is replaced by the [[velocity field]] {{math|'''u'''('''r''', ''t'')}} of a fluid, this is a term in the [[material derivative]] (with many other names) of [[continuum mechanics]], with another term being the partial time derivative:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>which could act on scalar or vector fields. In fact, if '''A''' is replaced by the [[velocity field]] {{math|'''u'''('''r''', ''t'')}} of a fluid, this is a term in the [[material derivative]] (with many other names) of [[continuum mechanics]], with another term being the partial <ins style="font-weight: bold; text-decoration: none;">[[</ins>time derivative<ins style="font-weight: bold; text-decoration: none;">]]</ins>:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block"> \frac{D}{D t} = \frac{\partial}{\partial t} + \mathbf{u}\cdot\nabla</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block"> \frac{D}{D t} = \frac{\partial}{\partial t} + \mathbf{u}\cdot\nabla</math></div></td></tr>
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imported>Localwerewolfenjoyer
http://debianws.lexgopc.com/wiki143/index.php?title=Cartesian_tensor&diff=1339396&oldid=prev
imported>OlliverWithDoubleL: Added short description
2024-10-27T20:42:40Z
<p>Added short description</p>
<a href="http://debianws.lexgopc.com/wiki143/index.php?title=Cartesian_tensor&diff=1339396">Show changes</a>
imported>OlliverWithDoubleL