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In mathematics, the **interior product** (also known as **interior derivative**, **interior multiplication**, **inner multiplication**, **inner derivative**, **insertion operator**, or **inner derivation**) is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold. The interior product, named in opposition to the exterior product, should not be confused with an inner product. The interior product is sometimes written as ^{[1]}

The interior product is defined to be the contraction of a differential form with a vector field. Thus if is a vector field on the manifold then

The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms

By antisymmetry of forms,

The interior product relates the exterior derivative and Lie derivative of differential forms by the **Cartan formula** (also known as the **Cartan identity**, **Cartan homotopy formula**^{[2]} or **Cartan magic formula**):

This identity defines a duality between the exterior and interior derivatives. Cartan's identity is important in symplectic geometry and general relativity: see moment map.^{[3]} The Cartan homotopy formula is named after Élie Cartan.^{[4]}

The interior product with respect to the commutator of two vector fields satisfies the identity

**^**The character ⨼ is U+2A3C in Unicode**^**Tu, Sec 20.5.**^**There is another formula called "Cartan formula". See Steenrod algebra.**^***Is "Cartan's magic formula" due to Élie or Henri?*, MathOverflow, 2010-09-21, retrieved 2018-06-25

- Theodore Frankel,
*The Geometry of Physics: An Introduction*; Cambridge University Press, 3rd ed. 2011 - Loring W. Tu,
*An Introduction to Manifolds*, 2e, Springer. 2011. doi:10.1007/978-1-4419-7400-6