The concept of areal differentiation

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The concept of areal differentiation

Specific vectors in a vector space[ edit ] Null vectorthe additive identity in a vector space. In a normed vector spaceit is the unique vector of norm zero.

In a Euclidean vector spaceit is the unique vector of length zero. Basis vector an element of a given basis of a vector space. Unit vectora vector in a normed vector space whose norm is 1, or a Euclidean vector of length one.

Vectors in specific vector spaces[ edit ] Column vectora matrix with only one column. The column vectors with a fixed number of rows form a vector space.

Row vectora matrix with only one row. The row vectors with a fixed number of rows form a vector space. Coordinate vectorthe n-uple of the coordinates of a vector on a basis of n elements.

Displacement vectora vector that specifies the change in position of a point relative to a previous position.

Areal differentiation - Oxford Reference

Displacement vectors belong to the vector space of translations. Position vector of a point, the displacement vector from a reference point called the origin to the point. A position vector represents the position of a point in a Euclidean space or an affine space.

Velocity vectorthe derivative, with respect of the time, of the position vector. It does not depend of the choice of the origin, and, thus belongs to the vector space of translations.

Pseudovectoralso called axial vector, an element of the dual of a vector space. In a inner product spacethe inner product defines an isomorphism between the space and its dual, which may make difficult to distinguish a pseudo vector from a vector. The distinction becomes apparent when one changes coordinates: Tangent vectoran element of the tangent space of a curvea surface or, more generally, a differential manifold at a given point these tangent spaces are naturally endowed with a structure of vector space Normal vector or simply normal, in a Euclidean space or, more generally, in an inner product space, a vector that is perpendicular to a tangent space at a point.

Normals are pseudovectors that belong to the dual of the tangent space. Gradientthe coordinates vector of the partial derivatives of a function of several real variables. In a Euclidean space the gradient gives the magnitude and direction of maximum increase of a scalar field. The gradient is a pseudo vector that is normal to a level curve.

When such tuples are used for representing some data, it is common to call them vectors even if the vector addition does not mean anything for these data, which may make the terminology confusing.

Similarly, some physical phenomena involve a direction and a magnitude. They are often represented by vectors, even if operations of vector spaces do not apply to them.

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Rotation vectora Euclidean vector whose direction is that of the axis of a rotation and magnitude is the angle of the rotation. Darboux vectorthe areal velocity vector of the Frenet frame of a space curve Burgers vectora vector that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice Laplace—Runge—Lenz vectora vector used chiefly to describe the shape and orientation of the orbit of an astronomical body around another Interval vectorin musical set theory, an array that expresses the intervallic content of a pitch-class set Poynting vectorin physics, a vector representing the energy flux density of an electromagnetic field Probability vectorin statistics, a vector with non-negative entries that sum to one.

The concept of areal differentiation

Random vector or multivariate random variablein statisticsa set of real -valued random variables that may be correlated. However, a random vector may also refer to a random variable that takes its values in a vector space.

Wave vectora representation of the local phase evolution of a wave Vectors in algebras[ edit ] Every algebra over a field is a vector space, but elements an algebra are generally not called vectors. However, in some cases, they are called vectors, mainly for historical reasons.

The concept of areal differentiation

Vector quaterniona quaternion with a zero real part Multivector or p-vectoran element of the exterior algebra of a vector space. Spinorsalso called spin vectors have been introduced for extending the notion of rotation vector.The office of Xaveer De Geyter Architects (XDGA) concentrates not only on architectural realizations, but also on urban projects not limited by one particular field of study.

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Geography - Geography’s early research agenda in Europe: Geography’s 19th-century research directions were set by a few influential individuals, although not all of them were even formally associated with the discipline.

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