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  2. Riemann sum - Wikipedia

    en.wikipedia.org/wiki/Riemann_sum

    A Riemann sum of over [,] with partition is defined as = = () ... A generalized midpoint rule formula, also known as the enhanced midpoint integration, ...

  3. List of formulas in Riemannian geometry - Wikipedia

    en.wikipedia.org/wiki/List_of_formulas_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:

  4. Trapezoidal rule - Wikipedia

    en.wikipedia.org/wiki/Trapezoidal_rule

    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 ...

  5. Riemann integral - Wikipedia

    en.wikipedia.org/wiki/Riemann_integral

    One popular restriction is the use of "left-hand" and "right-hand" Riemann sums. In a left-hand Riemann sum, t i = x i for all i, and in a right-hand Riemann sum, t i = x i + 1 for all i. Alone this restriction does not impose a problem: we can refine any partition in a way that makes it a left-hand or right-hand sum by subdividing it at each t i.

  6. Partition of an interval - Wikipedia

    en.wikipedia.org/wiki/Partition_of_an_interval

    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.

  7. Midpoint method - Wikipedia

    en.wikipedia.org/wiki/Midpoint_method

    The midpoint method computes + so that the red chord is approximately parallel to the tangent line at the midpoint (the green line). In numerical analysis , a branch of applied mathematics , the midpoint method is a one-step method for numerically solving the differential equation ,

  8. Calculus - Wikipedia

    en.wikipedia.org/wiki/Calculus

    A sequence of midpoint Riemann sums over a regular partition of an interval: the total area of the rectangles converges to the integral of the function. Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral .

  9. Gauss circle problem - Wikipedia

    en.wikipedia.org/wiki/Gauss_circle_problem

    The value of () can be given by several series. In terms of a sum involving the floor function it can be expressed as: [5] = + = (⌊ + ⌋ ⌊ + ⌋).This is a consequence of Jacobi's two-square theorem, which follows almost immediately from the Jacobi triple product.