Conormal

A term used in the theory of boundary value problems for partial differential equations (cf. Boundary value problem, partial differential equations). Let $ \pmb\nu = ( \nu _ {1} \dots \nu _ {n} ) $ be the outward normal at a point $ x $ to a smooth surface $ S $ situated in a Euclidean space $ E ^ {n} $ with coordinates $ x ^ {1} \dots x ^ {n} $, and let $ g ^ {ij} $ be a contravariant continuous tensor, usually representing the coefficients of some second-order (elliptic) differential operator $ D = g ^ {ij} ( \partial / \partial x ^ {i} ) ( \partial / \partial x ^ {j} ) $. Then the conormal (with respect to $ D $) is the vector

$$ \mathbf n = \ ( \nu ^ {1} \dots \nu ^ {n} ), $$

where $ \nu ^ {i} = g ^ {ik} \nu _ {k} $. In other words, the conormal is the contravariant description (in the space with metric defined by the tensor inverse to $ g ^ {ij} $) of the normal covariant vector $ \pmb\nu $ to $ S $ (in the space with Euclidean metric).

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