analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domains...

length) and direction. Euclidean vectors can be added and scaled to form a vector space. A vector quantity is a vector-valued physical quantity, including...

producing a vector v(t) as the result. In terms of the standard unit vectors i, j, k of Cartesian 3-space, these specific types of vector-valued functions are...

mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X {\displaystyle...

L^{p}} spaces have been defined for such functions. Differentiation in Fréchet spaces Differentiable vector–valued functions from Euclidean space – Differentiable...

holomorphic functions on an open domain, spaces of infinitely differentiable functions, the Schwartz spaces, and spaces of test functions and the spaces of distributions...

on the above sorts of vectors. A vector space formed by geometric vectors is called a Euclidean vector space, and a vector space formed by tuples is called...

fields, primarily in three-dimensional Euclidean space, R 3 . {\displaystyle \mathbb {R} ^{3}.} The term vector calculus is sometimes used as a synonym...

mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one...

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional...

In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space R n {\displaystyle...

analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences...

of the vector space. Similarly, the Cartesian coordinates of the points of a Euclidean space of dimension n, En (Euclidean line, E; Euclidean plane, E2;...

engineering, test functions are usually infinitely differentiable complex-valued (or real-valued) functions with compact support that are defined on some given...

numbers to vectors in some vector space R n {\displaystyle \mathbb {R} ^{n}} . A vector-valued function can be split up into its coordinate functions y 1 (...

absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space. A real-valued function on a...

operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols ∇ ⋅ ∇ {\displaystyle \nabla...

Scalars can also be, more generally, elements of any field. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities (such as...

functions from Euclidean space – Differentiable function in functional analysis Infinite-dimensional vector function – function whose values lie in an infinite-dimensional...

targets Differentiable vector-valued functions from Euclidean space – Differentiable function in functional analysis Differentiation in Fréchet spaces Fractal...

Cr-parametrization is a vector-valued function γ : I → R n {\displaystyle \gamma :I\to \mathbb {R} ^{n}} that is r-times continuously differentiable (that is, the...

function is differentiable just once on an open set, it is both infinitely differentiable and analytic on that set.[citation needed] Smooth functions...

define implicit functions, namely those that are obtained by equating to zero multivariable functions that are continuously differentiable. A common type...

Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are...

nonnegative values of the variable, and not differentiable at 0 (it is differentiable for all positive values of the variable). A real-valued function of a real...

analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples...

made there apply to all vector bundles). Let M be a differentiable manifold, such as Euclidean space. A vector-valued function M → R n {\displaystyle M\to...

}{|\mathbf {v} |}}.} In the context of a function on a Euclidean space, some texts restrict the vector v to being a unit vector. With this restriction, both the...

analysis, a Banach space (/ˈbɑː.nʌx/, Polish pronunciation: [ˈba.nax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric...

of three-dimensional Euclidean space are common. Assume that (x, y, z) is a given Cartesian coordinate system, that A is a vector field with components...