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Directional Derivative
In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. The directional derivative of a scalar function ''f'' with respect to a vector v at a point (e.g., position) x may be denoted by any of the following: \nabla_(\mathbf)=f'_\mathbf(\mathbf)=D_\mathbff(\mathbf)=Df(\mathbf)(\mathbf)=\partial_\mathbff(\mathbf)=\mathbf\cdot=\mathbf\cdot \frac. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant. The directional derivative is a special case of the Gateaux derivative. Definition The ''directional derivative'' of a scalar function :f(\mathbf) = f(x_1, x_2, \ldots, x_n) along a vector :\mathbf = (v_1, \ldots, v_n) is the function \nabla_ defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point , the direction of the gradient is the direction in which the function increases most quickly from , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent. In coordinate-free terms, the gradient of a function f(\bf) may be defined by: :df=\nabla f \cdot d\bf where ''df'' is the total infinitesimal change in ''f'' for an infinitesimal displacement d\bf, and is seen to be maximal when d\bf is in the direction of the gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of germs. Formally, a tangent vector at the point x is a linear derivation of the algebra defined by the set of germs at x. Motivation Before proceeding to a general definition of the tangent vector, we discuss its use in calculus and its tensor properties. Calculus Let \mathbf(t) be a parametric smooth curve. The tangent vector is given by \mathbf'(t), where we have used a prime instead of the usual dot to indicate differentiation with respect to parameter . The unit tangent vector is given by \mathbf(t) = \frac\,. Example Given the curve \mathbf(t) = \left\ in \R^3, the unit tangent vector at t = 0 is given by \mathbf(0) = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chain Rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , then the chain rule is, in Lagrange's notation, :h'(x) = f'(g(x)) g'(x). or, equivalently, :h'=(f\circ g)'=(f'\circ g)\cdot g'. The chain rule may also be expressed in Leibniz's notation. If a variable depends on the variable , which itself depends on the variable (that is, and are dependent variables), then depends on as well, via the intermediate variable . In this case, the chain rule is expressed as :\frac = \frac \cdot \frac, and : \left.\frac\_ = \left.\frac\_ \cdot \left. \frac\_ , for indicating at which points the derivatives have to be evaluated. In integration, the counterpart to the chain rule is the substitution rule. Intuitive explanation Intuitively, the chain rule states that knowing the instantaneous rate of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + u \cdot v' or in Leibniz's notation as \frac (u\cdot v) = \frac \cdot v + u \cdot \frac. The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts. Discovery Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. (However, J. M. Child, a translator of Leibniz's papers, argues that it is due to Isaac Barrow.) Here is Leibniz's argument: Let ''u''(''x'') and ''v''(''x'') be two differentiable functions of ''x''. Then the differential of ''uv'' is : \begin d(u\cdot v) & = (u + du)\cdot (v + dv) - u\cdot v \\ & = u\cdot dv + v\cdot du + du\cdot dv. \end Since the term ''du''·''dv'' is "negli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant Factor Rule In Differentiation
This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. Elementary rules of differentiation Unless otherwise stated, all functions are functions of real numbers (R) that return real values; although more generally, the formulae below apply wherever they are well defined — including the case of complex numbers (C). Constant term rule For any value of c, where c \in \mathbb, if f(x) is the constant function given by f(x) = c, then \frac = 0. Proof Let c \in \mathbb and f(x) = c. By the definition of the derivative, :\begin f'(x) &= \lim_\frac \\ &= \lim_ \frac \\ &= \lim_ \frac \\ &= \lim_ 0 \\ &= 0 \end This shows that the derivative of any constant function is 0. Differentiation is linear For any functions f and g and any real numbers a and b, the derivative of the function h(x) = af(x) + bg(x) with respect to x is: h'(x) = a f'(x) + b g'(x). In Leibniz's notation this is written as: \frac = a\frac +b\fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum Rule In Differentiation
This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. Elementary rules of differentiation Unless otherwise stated, all functions are functions of real numbers (R) that return real values; although more generally, the formulae below apply wherever they are well defined — including the case of complex numbers (C). Constant term rule For any value of c, where c \in \mathbb, if f(x) is the constant function given by f(x) = c, then \frac = 0. Proof Let c \in \mathbb and f(x) = c. By the definition of the derivative, :\begin f'(x) &= \lim_\frac \\ &= \lim_ \frac \\ &= \lim_ \frac \\ &= \lim_ 0 \\ &= 0 \end This shows that the derivative of any constant function is 0. Differentiation is linear For any functions f and g and any real numbers a and b, the derivative of the function h(x) = af(x) + bg(x) with respect to x is: h'(x) = a f'(x) + b g'(x). In Leibniz's notation this is written as: \frac = a\frac +b\fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Derivative
In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as considering all partial derivatives simultaneously. The term "total derivative" is primarily used when is a function of several variables, because when is a function of a single variable, the total derivative is the same as the ordinary derivative of the function. The total derivative as a linear map Let U \subseteq \R^n be an open subset. Then a function f:U \to \R^m is said to be (totally) differentiable at a point a\in U if there exists a linear transformation df_a:\R^n \to \R^m such that :\lim_ \frac=0. The linear map df_a is called the (total) derivative or (total) differential of f at a. Other notations for the total derivative inclu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighborhood (mathematics)
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set. Definitions Neighbourhood of a point If X is a topological space and p is a point in X, then a of p is a subset V of X that includes an open set U containing p, p \in U \subseteq V \subseteq X. This is also equivalent to the point p \in X belonging to the topological interior of V in X. The neighbourhood V need be an open subset X, but when V is open in X then it is called an . Some authors have been known to require neighbourhoods to be open, so it is important to note conventions. A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vector'', commonly denoted as d, is used to describe a unit vector being used to represent spatial direction and relative direction. 2D spatial directions are numerically equivalent to points on the unit circle and spatial directions in 3D are equivalent to a point on the unit sphere. The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e., :\mathbf = \frac where , u, is the norm (or length) of u. The term ''normalized vector'' is sometimes used as a synonym for ''unit vector''. Unit vectors are often chosen to form the basis of a vector space, and every vector in the space may be written as a linear combination of unit vectors. Orthogonal coordinates Cartesian coordinates Unit vectors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normalized Vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vector'', commonly denoted as d, is used to describe a unit vector being used to represent spatial direction and relative direction. 2D spatial directions are numerically equivalent to points on the unit circle and spatial directions in 3D are equivalent to a point on the unit sphere. The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e., :\mathbf = \frac where , u, is the norm (or length) of u. The term ''normalized vector'' is sometimes used as a synonym for ''unit vector''. Unit vectors are often chosen to form the basis of a vector space, and every vector in the space may be written as a linear combination of unit vectors. Orthogonal coordinates Cartesian coordinates Unit vectors may be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the '' Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
