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).

References

[1]	A.V. Bitsadze, "Equations of mathematical physics" , MIR (1980) (Translated from Russian)
[2]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)

References

[a1]	A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964)