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This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. Derivatives to nth order. Some rules exist for computing the n-th derivative of functions, where n is a positive integer. These include: Faà di Bruno's formula
List of mathematical series. This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. is a Bernoulli polynomial. is an Euler number. is the Riemann zeta function. is the gamma function. is a polygamma function. is a polylogarithm.
Indefinite sum. In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by or , [1] [2] is the linear operator, inverse of the forward difference operator . It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus. More explicitly, if , then.
t. e. The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). The two operations are inverses of each other ...
The exterior derivative is: The last formula, where summation starts at i = 3, follows easily from the properties of the exterior product. Namely, dxi ∧ dxi = 0 . Example 2. Let σ = u dx + v dy be a 1 -form defined over ℝ2. By applying the above formula to each term (consider x1 = x and x2 = y) we have the sum.
A finite difference is a mathematical expression of the form f (x + b) − f (x + a).If a finite difference is divided by b − a, one gets a difference quotient.The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.
v. t. e. In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity. [1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such ...
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus.