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Loosely speaking, a function is Riemann integrable if all Riemann sums converge as the partition "gets finer and finer". While not derived as a Riemann sum, taking the average of the left and right Riemann sums is the trapezoidal rule and gives a trapezoidal sum. It is one of the simplest of a very general way of approximating integrals using ...
A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with the norm of the partition indicated in red. In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x 0, x 1, x 2, …, x n of real numbers such that a = x 0 < x 1 < x 2 < … < x n = b.
The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" (or "composite ...
Its Riemann sum is within ε of s, and any refinement of this partition will also have mesh less than δ, so the Riemann sum of the refinement will also be within ε of s. To show that the second definition implies the first, it is easiest to use the Darboux integral.
A Riemann sum of a function f with respect to such a tagged partition is defined as ∑ i = 1 n f ( t i ) Δ i ; {\displaystyle \sum _{i=1}^{n}f(t_{i})\,\Delta _{i};} thus each term of the sum is the area of a rectangle with height equal to the function value at the chosen point of the given sub-interval, and width the same as the width of sub ...
Roughly speaking, one chooses a sequence of partitions of the interval from 0 to t and constructs Riemann sums. Every time we are computing a Riemann sum, we are using a particular instantiation of the integrator. It is crucial which point in each of the small intervals is used to compute the value of the function. The limit then is taken in ...
The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero: = = = = The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors:
The Riemann–Stieltjes integral can be efficiently handled using an appropriate generalization of Darboux sums. For a partition P and a nondecreasing function g on [a, b] define the upper Darboux sum of f with respect to g by