Cartan formula

Do not confuse with Cartan lemma.

Also called Cartan identity, Cartan homotopy formula or Cartan magic formula

L_{X} = i_{X} \circ d + d \circ i_{X}

where for a $p$ -form $ω$ is the $p - 1$ -form$$
i_X(\omega)(Y_1,Y_2, \ldots)=\omega(X,Y_1,Y_2, \ldots)

I f w e t a k e a s a d e f i n i t i o n

i_X(f)=0

f o r a f u n c t i o n $ f $ t h e n w e h a v e t h e k n o w f a c t :

\mathcal{L}{\mathbf{X}}(f)=i{\mathbf{X}}(df)

S y m b o l i c a l l y i t c a n b e d e n o t e d

\left{d, \iota_{X}\right} \omega:=\mathcal{L}{X} \omega=d\left(\iota \omega\right)+\iota_{X} d \omega.

# Other useful formulas Seeformulas for Lie derivative, exterior derivatives, bracket, interior product