[include(틀:해석학·미적분학)]
[목차]
== 개요 ==
야코비 공식( 또는 자코비 공식,Jacobi formula)는  [[벡터 미적분학]](vector calculus) 및  [[행렬 미적분학]](matrix calculus)등에서 사용하는 행렬(matrix)의 [[미분]](differential, differentiation)을 다룰수있는 주요하게 관련된 공식이다.

== 자코비 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) )][* (physics,stackexchange)Variation of determinant of the metric tensor [[https://physics.stackexchange.com/questions/218486/variation-of-determinant-of-the-metric-tensor]]][* (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]]]
[math(\rm{det()} )]는 [[행렬식]] , [math( \rm{tr()} )]는 [[주대각합]], [math(\square^{-1})]는 [[역행렬]] , [math( \rm{adj()}  )]는 [[딸림행렬]]이다.
== 행렬미분 예 ==
=== [[편미분]] ===
[math( \partial \rm{det} M = \rm{det}M \;tr \left( M^{-1} \partial M\right) )]
=== [[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} )]
==== 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) )]
== 관련 문서 ==
 * [[자코비 행렬]]
 * [[라그랑지 승수법]]
 * [[힐베르트 액션]]
[[분류: 미적분 ]]