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The product-to-sum identities [28] or prosthaphaeresis formulae can be proven by expanding their right-hand sides using the angle addition theorems. Historically, the first four of these were known as Werner's formulas , after Johannes Werner who used them for astronomical calculations. [ 29 ]
However, if the terms and their finite sums belong to a set that has limits, it may be possible to assign a value to a series, called the sum of the series. This value is the limit as n {\displaystyle n} tends to infinity of the finite sums of the n {\displaystyle n} first terms of the series if the limit exists.
A product integral is any product-based counterpart of the usual sum-based integral of calculus. The product integral was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations .
Identity 1: + = The following two results follow from this and the ratio identities. To obtain the first, divide both sides of + = by ; for the second, divide by .
The sum of the series is a random variable whose probability density function is close to for values between and , and decreases to near-zero for values greater than or less than . Intermediate between these ranges, at the values ± 2 {\displaystyle \pm 2} , the probability density is 1 8 − ε {\displaystyle {\tfrac {1}{8}}-\varepsilon } for ...
Integration by parts can be extended to functions of several variables by applying a version of the fundamental theorem of calculus to an appropriate product rule. There are several such pairings possible in multivariate calculus, involving a scalar-valued function u and vector-valued function (vector field) V. [7]
denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold . , denote the tangent spaces of , at the points , , respectively. denotes the cotangent space of at the point .
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
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