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Multi-index Notation
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. Definition and basic properties An ''n''-dimensional multi-index is an ''n''-tuple :\alpha = (\alpha_1, \alpha_2,\ldots,\alpha_n) of non-negative integers (i.e. an element of the ''n''-dimensional set of natural numbers, denoted \mathbb^n_0). For multi-indices \alpha, \beta \in \mathbb^n_0 and x = (x_1, x_2, \ldots, x_n) \in \mathbb^n one defines: ;Componentwise sum and difference :\alpha \pm \beta= (\alpha_1 \pm \beta_1,\,\alpha_2 \pm \beta_2, \ldots, \,\alpha_n \pm \beta_n) ;Partial order :\alpha \le \beta \quad \Leftrightarrow \quad \alpha_i \le \beta_i \quad \forall\,i\in\ ;Sum of components (absolute value) :, \alpha , = \alpha_1 + \alpha_2 + \cdots + \alpha_n ;Factorial :\alpha ! = \alpha_1! \cdot \alpha_2! \cdots \alpha_n! ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Notation
Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathematics, science, and engineering for representing complex concepts and properties in a concise, unambiguous and accurate way. For example, Albert Einstein's equation E=mc^2 is the quantitative representation in mathematical notation of the mass–energy equivalence. Mathematical notation was first introduced by François Viète at the end of the 16th century, and largely expanded during the 17th and 18th century by René Descartes, Isaac Newton, Gottfried Wilhelm Leibniz, and overall Leonhard Euler. Symbols The use of many symbols is the basis of mathematical notation. They play a similar role as words in natural languages. They may play different roles in mathematical notation similarly as verbs, adjective and nouns play different r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multinomial Theorem
In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. Theorem For any positive integer and any non-negative integer , the multinomial formula describes how a sum with terms expands when raised to an arbitrary power : :(x_1 + x_2 + \cdots + x_m)^n = \sum_ \prod_^m x_t^\,, where : = \frac is a multinomial coefficient. The sum is taken over all combinations of nonnegative integer indices through such that the sum of all is . That is, for each term in the expansion, the exponents of the must add up to . Also, as with the binomial theorem, quantities of the form that appear are taken to equal 1 ( even when equals zero). In the case , this statement reduces to that of the binomial theorem. Example The third power of the trinomial is given by :(a+b+c)^3 = a^3 + b^3 + c^3 + 3 a^2 b + 3 a^2 c + 3 b^2 a + 3 b^2 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Index Notation
In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to the elements of a list, a vector, or a matrix, depending on whether one is writing a formal mathematical paper for publication, or when one is writing a computer program. In mathematics It is frequently helpful in mathematics to refer to the elements of an array using subscripts. The subscripts can be integers or variables. The array takes the form of tensors in general, since these can be treated as multi-dimensional arrays. Special (and more familiar) cases are vectors (1d arrays) and matrices (2d arrays). The following is only an introduction to the concept: index notation is used in more detail in mathematics (particularly in the representation and manipulation of tensor operations). See the main article for further details. One- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein Notation
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916. Introduction Statement of convention According to this convention, when an index variable appears twice in a single term and is not otherwise defined (see Free and bound variables), it implies summation of that term over all the values of the index. So where the indices can range over the set , : y = \sum_^3 c_i x^i = c_1 x^1 + c_2 x^2 + c_3 x^3 is simplified by the convention to: : y = c_i x^i The upper indices are not exponents but are indices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point. Differential calculus and integral calculu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Rule
In calculus, the power rule is used to differentiate functions of the form f(x) = x^r, whenever r is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule. The power rule underlies the Taylor series as it relates a power series with a function's derivatives. Statement of the power rule Let f be a function satisfying f(x)=x^r for all x, where r \in \mathbb. Then, :f'(x) = rx^ \, . The power rule for integration states that :\int\! x^r \, dx=\frac+C for any real number r \neq -1. It can be derived by inverting the power rule for differentiation. In this equation C is any constant. Proofs Proof for real exponents To start, we should choose a working definition of the value of f(x) = x^r, where r is any real number. Although it is feasible to define the value as the limit of a sequence of rational powers that approach the irrational power whenever we encounter such a power, or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (''strong derivative'') for functions not assumed differentiable, but only integrable, i.e., to lie in the L''p'' space L^1( ,b. The method of integration by parts holds that for differentiable functions u and \varphi we have :\begin \int_a^b u(x) \varphi'(x) \, dx & = \Big (x) \varphi(x)\Biga^b - \int_a^b u'(x) \varphi(x) \, dx. \\ pt \end A function ''u''' being the weak derivative of ''u'' is essentially defined by the requirement that this equation must hold for all infinitely differentiable functions ''φ'' vanishing at the boundary points (\varphi(a)=\varphi(b)=0). Definition Let u be a function in the Lebesgue space L^1( ,b. We say that v in L^1( ,b is a weak derivative of u if :\int_a^b u(t)\varphi'(t) \, dt=-\int_a^b v(t)\varphi(t) \, dt for ''all'' infinitely differentiable functions \varphi with \varphi(a)=\varphi(b)=0. Generalizing to n dimensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distribution (mathematics)
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function. A function f is normally thought of as on the in the function domain by "sending" a point x in its domain to the point f(x). Instead of acting on points, distribution theory reinterp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Support
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis. Formulation Suppose that f : X \to \R is a real-valued function whose domain is an arbitrary set X. The of f, written \operatorname(f), is the set of points in X where f is non-zero: \operatorname(f) = \. The support of f is the smallest subset of X with the property that f is zero on the subset's complement. If f(x) = 0 for all but a finite number of points x \in X, then f is said to have . If the set X has an additional structure (for example, a topology), then the support of f is defined in an analogous way as the smallest subset of X of an appropriate type such that f vanishes in an appropriate sense on its complement. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integration By Parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation. The integration by parts formula states: \begin \int_a^b u(x) v'(x) \, dx & = \Big (x) v(x)\Biga^b - \int_a^b u'(x) v(x) \, dx\\ & = u(b) v(b) - u(a) v(a) - \int_a^b u'(x) v(x) \, dx. \end Or, letting u = u(x) and du = u'(x) \,dx while v = v(x) and dv = v'(x) \, dx, the formula can be written more compactly: \int u \, dv \ =\ uv - \int v \, du. Mathematician Brook Taylor discovered integration by parts, first publishing the idea in 1715. More general formulations of integration by parts ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative. Definition An order-m linear differential operator is a map A from a function space \mathcal_1 to another function space \mathcal_2 that can be written as: A = \sum_a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative integers, , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n, and for each \alpha, a_\alpha(x) is a function on some open domain in ''n''-dimensional space. The operator D^\alpha is interpreted as D^\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about ''x''0 converges to the function in some neighborhood for every ''x''0 in its domain. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write : f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots in which the coefficients a_0, a_1, \dots are real numbers and the series is convergent to f(x) for x in a neighborhood of x_0. Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point x_0 in its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
