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Display information for equation id:math.2497.41 on revision:2497
* Page found: Thermodynamische Stabilität (eq math.2497.41)
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Hash: 710bbdf63b50ca780c05c42213c19a05
TeX (original user input):
\begin{align}
& \left\langle \Delta {{M}^{\nu }}\Delta {{M}^{\mu }} \right\rangle =-\frac{\partial \left\langle {{M}^{\mu }} \right\rangle }{\partial {{\lambda }_{\nu }}}=-\frac{\partial \left\langle {{M}^{\nu }} \right\rangle }{\partial {{\lambda }_{\mu }}} \\
& \Rightarrow \left\langle {{\left( \Delta b \right)}^{2}} \right\rangle =-{{\lambda }_{\nu }}{{\lambda }_{\mu }}\frac{\partial \left\langle {{M}^{\nu }} \right\rangle }{\partial {{\lambda }_{\mu }}}=-\frac{1}{k}\frac{\partial S}{\partial {{\lambda }_{\mu }}}{{\lambda }_{\mu }} \\
\end{align}
TeX (checked):
{\begin{aligned}&\left\langle \Delta {{M}^{\nu }}\Delta {{M}^{\mu }}\right\rangle =-{\frac {\partial \left\langle {{M}^{\mu }}\right\rangle }{\partial {{\lambda }_{\nu }}}}=-{\frac {\partial \left\langle {{M}^{\nu }}\right\rangle }{\partial {{\lambda }_{\mu }}}}\\&\Rightarrow \left\langle {{\left(\Delta b\right)}^{2}}\right\rangle =-{{\lambda }_{\nu }}{{\lambda }_{\mu }}{\frac {\partial \left\langle {{M}^{\nu }}\right\rangle }{\partial {{\lambda }_{\mu }}}}=-{\frac {1}{k}}{\frac {\partial S}{\partial {{\lambda }_{\mu }}}}{{\lambda }_{\mu }}\\\end{aligned}}
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<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><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi mathvariant="normal">Δ</mi><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msup><mi mathvariant="normal">Δ</mi><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>μ</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo>=</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>μ</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>=</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>μ</mi></mrow></msub></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi mathvariant="normal">Δ</mi><mi>b</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo>=</mo><mo>−</mo><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msub><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>μ</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>μ</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>=</mo><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi>k</mi></mrow></mfrac></mrow><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><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>μ</mi></mrow></msub></mrow></mrow></mfrac></mrow><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mi>μ</mi></mrow></msub></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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