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The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on ...
Complex analysis. In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula.
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space ...
A form of the mean value theorem, where a < ξ < b, can be applied to the first and last integrals of the formula for Δ φ above, resulting in. Dividing by Δ α, letting Δ α → 0, noticing ξ1 → a and ξ2 → b and using the above derivation for yields. This is the general form of the Leibniz integral rule.
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat ), is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if is holomorphic in a simply connected domain Ω, then ...
Buffon's needle was the earliest problem in geometric probability to be solved; [2] it can be solved using integral geometry. The solution for the sought probability p, in the case where the needle length l is not greater than the width t of the strips, is. This can be used to design a Monte Carlo method for approximating the number π ...
Image of the Large Magellanic Cloud, one of the nearest galaxies to our Milky Way, created with LIC. In scientific visualization, line integral convolution ( LIC) is a method to visualize a vector field (such as fluid motion) at high spatial resolutions. [1] The LIC technique was first proposed by Brian Cabral and Leith Casey Leedom in 1993.
The integral form of the original circuital law is a line integral of the magnetic field around some closed curve C (arbitrary but must be closed). The curve C in turn bounds both a surface S which the electric current passes through (again arbitrary but not closed—since no three-dimensional volume is enclosed by S ), and encloses the current.
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