1. 개요
Jacobi formula / 야코비 公 式 / (독일어)Jacobis Formel벡터 미적분학 및 행렬 미적분학 등에서 사용하는 행렬의 미분을 다룰 수 있게 하는 공식이다.
2. 야코비 공식
[math( \it{d} \;\rm{M} =\it{d} \;\rm{det} \;M = \rm{det} \;M \;tr\left( M^{-1} \; \it{d}M\right) = \;tr\left( adj\;M \; \it{d}M\right) )][1][2][math(\rm{det()} )]는 행렬식 , [math( \rm{tr()} )]는 주대각합, [math(\square^{-1})]는 역행렬 , [math( \rm{adj()})]는 딸림행렬이다.
3. 행렬미분 예
3.1. 편미분
[math( \partial \rm{det} M = \rm{det}M \;tr \left( M^{-1} \partial M\right) )]3.2. 변분
[math( \delta g = \delta det(g_{ab}) = g\left( g^{ab}\delta g_{ab} \right) )][math( -g^{ab}\delta g_{ab} = -\left(-g^{ab}\delta g_{ab}\right) = g_{ab}\delta g^{ab} )]
3.2.1. 예
[math( \delta \sqrt{-g} = -\dfrac{1}{2}\dfrac{1}{ \sqrt{-g} } \delta g = -\dfrac{1}{2}\dfrac{1}{ \sqrt{-g} } g\left( g^{ab}\delta g_{ab} \right) = -\dfrac{1}{2}\dfrac{g}{ \sqrt{-g} } \left( g^{ab}\delta g_{ab} \right) )]
[math( = -\dfrac{1}{2}\dfrac{g \sqrt{-g}}{ \sqrt{-g} \sqrt{-g} } \left( g^{ab}\delta g_{ab} \right) = -\dfrac{1}{2}\dfrac{g \sqrt{-g}}{ -g } \left( g^{ab}\delta g_{ab} \right) = -\dfrac{1}{2}\dfrac{-g \sqrt{-g}}{ -g } \left( g_{ab}\delta g^{ab} \right) = -\dfrac{1}{2}\sqrt{-g}\left( g_{ab}\delta g^{ab} \right) )]
[math( \delta \sqrt{g} = \dfrac{1}{2}\dfrac{1}{ \sqrt{g} } \delta g = \dfrac{1}{2}\dfrac{1}{ \sqrt{g} } g\left( g^{ab}\delta g_{ab} \right) = \dfrac{1}{2}\dfrac{g}{ \sqrt{g} } \left( g^{ab}\delta g_{ab} \right) )]
[math( = \dfrac{1}{2}\dfrac{g\sqrt{g}}{ \sqrt{g}\sqrt{g} } \left( g^{ab}\delta g_{ab} \right) = \dfrac{1}{2}\dfrac{g\sqrt{g}}{ g } \left( g^{ab}\delta g_{ab} \right) = \dfrac{1}{2} \sqrt{g} \left( g^{ab}\delta g_{ab} \right) = -\dfrac{1}{2} \sqrt{g} \left( g_{ab}\delta g^{ab} \right) )]
4. 관련 문서
[1]
(physics,stackexchange)Variation of determinant of the metric tensor
https://physics.stackexchange.com/questions/218486/variation-of-determinant-of-the-metric-tensor
[2]
(Berkeley EECS)Math. H110 Jacobi’s Formula for d det(B) October 26, 1998 3:53 am Prof. W. Kahan Page 1/4 Jacobi's Formula for the Derivative of a Determinant
https://people.eecs.berkeley.edu/~wkahan/MathH110/jacobi.pdf