Vector notationIn mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more generally, members of a vector space. For representing a vector, the common typographic convention is lower case, upright boldface type, as in v. The International Organization for Standardization (ISO) recommends either bold italic serif, as in v, or non-bold italic serif accented by a right arrow, as in . In advanced mathematics, vectors are often represented in a simple italic type, like any variable.
Del in cylindrical and spherical coordinatesThis is a list of some vector calculus formulae for working with common curvilinear coordinate systems. This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): The polar angle is denoted by : it is the angle between the z-axis and the radial vector connecting the origin to the point in question. The azimuthal angle is denoted by : it is the angle between the x-axis and the projection of the radial vector onto the xy-plane.
Approximation affineEn mathématiques, une approximation affine est une approximation d'une fonction au voisinage d'un point à l'aide d'une fonction affine. Une approximation affine sert principalement à simplifier un problème dont on peut obtenir une solution approchée. Deux façons classiques d'obtenir une approximation affine de fonction passent par l'interpolation ou le développement limité à l’ordre 1.
Skew coordinatesA system of skew coordinates is a curvilinear coordinate system where the coordinate surfaces are not orthogonal, in contrast to orthogonal coordinates. Skew coordinates tend to be more complicated to work with compared to orthogonal coordinates since the metric tensor will have nonzero off-diagonal components, preventing many simplifications in formulas for tensor algebra and tensor calculus. The nonzero off-diagonal components of the metric tensor are a direct result of the non-orthogonality of the basis vectors of the coordinates, since by definition: where is the metric tensor and the (covariant) basis vectors.
Fréchet derivativeIn mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations. Generally, it extends the idea of the derivative from real-valued functions of one real variable to functions on normed spaces.
Semi-differentiabilityIn calculus, a branch of mathematics, the notions of one-sided differentiability and semi-differentiability of a real-valued function f of a real variable are weaker than differentiability. Specifically, the function f is said to be right differentiable at a point a if, roughly speaking, a derivative can be defined as the function's argument x moves to a from the right, and left differentiable at a if the derivative can be defined as x moves to a from the left.
Isomorphisme musicalEn mathématiques, plus précisément en géométrie différentielle, l'isomorphisme musical (ou isomorphisme canonique ) est un isomorphisme entre le fibré tangent et le fibré cotangent d'une variété pseudo-riemannienne induite par son tenseur métrique. Il existe des isomorphismes similaires sur les variétés symplectiques. Le terme musical fait référence à l'utilisation des symboles (bémol) et (dièse). En notation covariante et contravariante, il est également connu sous le nom d'indice d'élévation et d'abaissement.
