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A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain could be 1 or greater than 1); the ...
Convex combination of two functions as vectors in a vector space of functions - visualized in Open Source Geogebra with [,] = [,] and as the first function : [,] a polynomial is defined. f ( x ) := 3 10 ⋅ x 2 − 2 {\displaystyle f(x):={\frac {3}{10}}\cdot x^{2}-2} A trigonometric function g : [ a , b ] → R {\displaystyle g:[a,b]\to \mathbb ...
One can define the pullback of vector-valued forms by smooth maps just as for ordinary forms. The pullback of an E-valued form on N by a smooth map φ : M → N is an (φ*E)-valued form on M, where φ*E is the pullback bundle of E by φ. The formula is given just as in the ordinary case. For any E-valued p-form ω on N the pullback φ*ω is ...
Given a subset S of R n, a vector field is represented by a vector-valued function V: S → R n in standard Cartesian coordinates (x 1, …, x n). If each component of V is continuous, then V is a continuous vector field. It is common to focus on smooth vector fields, meaning that each component is a smooth function (differentiable any number ...
In vector calculus, the Jacobian matrix (/ dʒ ə ˈ k oʊ b i ə n /, [1] [2] [3] / dʒ ɪ-, j ɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives.
The Jacobian matrix is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. [9] [10] A further generalization for a function between Banach spaces is the Fréchet derivative.
Let f(v) be a vector valued function of the vector v.Then the derivative of f(v) with respect to v (or at v) is the second order tensor defined through its dot product with any vector u being
In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being the values of the two state variables, say (x, y), or (q, p) etc. (any pair of variables).