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Display information for equation id:math.2446.12 on revision:2446
* Page found: Thermodynamischer Limes (eq math.2446.12)
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Hash: 8279aed079266bef0e7d83f1f59cfaa2
TeX (original user input):
\begin{align}
& S\left( U,V,{{{\bar{N}}}^{\alpha }} \right)=\frac{\partial S}{\partial U}U+\frac{\partial S}{\partial V}V+\frac{\partial S}{\partial {{{\bar{N}}}^{\alpha }}}{{{\bar{N}}}^{\alpha }}=\frac{1}{T}U+\frac{p}{T}V-\frac{{{\mu }_{\alpha }}}{T}{{{\bar{N}}}^{\alpha }} \\
& \frac{\partial S}{\partial U}=\frac{1}{T} \\
& \frac{\partial S}{\partial V}=\frac{p}{T} \\
& \frac{\partial S}{\partial {{{\bar{N}}}^{\alpha }}}=-\frac{{{\mu }_{\alpha }}}{T} \\
\end{align}
TeX (checked):
{\begin{aligned}&S\left(U,V,{{\bar {N}}^{\alpha }}\right)={\frac {\partial S}{\partial U}}U+{\frac {\partial S}{\partial V}}V+{\frac {\partial S}{\partial {{\bar {N}}^{\alpha }}}}{{\bar {N}}^{\alpha }}={\frac {1}{T}}U+{\frac {p}{T}}V-{\frac {{\mu }_{\alpha }}{T}}{{\bar {N}}^{\alpha }}\\&{\frac {\partial S}{\partial U}}={\frac {1}{T}}\\&{\frac {\partial S}{\partial V}}={\frac {p}{T}}\\&{\frac {\partial S}{\partial {{\bar {N}}^{\alpha }}}}=-{\frac {{\mu }_{\alpha }}{T}}\\\end{aligned}}
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MathML (4.183 KB / 472 B) :
<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><mi>S</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>U</mi><mo>,</mo><mi>V</mi><mo>,</mo><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>N</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>α</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>S</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>U</mi></mrow></mrow></mfrac></mrow><mi>U</mi><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>S</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>V</mi></mrow></mrow></mfrac></mrow><mi>V</mi><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>S</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>N</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>α</mi></mrow></msup></mrow></mrow></mfrac></mrow><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>N</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>α</mi></mrow></msup><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></mfrac></mrow><mi>U</mi><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>p</mi></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></mfrac></mrow><mi>V</mi><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msub><mi>μ</mi><mrow data-mjx-texclass="ORD"><mi>α</mi></mrow></msub></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></mfrac></mrow><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>N</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>α</mi></mrow></msup></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>S</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>U</mi></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>S</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>V</mi></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>p</mi></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>S</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>N</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>α</mi></mrow></msup></mrow></mrow></mfrac></mrow><mo>=</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msub><mi>μ</mi><mrow data-mjx-texclass="ORD"><mi>α</mi></mrow></msub></mrow><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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