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If D is a simple type of region with its boundary consisting of the curves C 1, C 2, C 3, C 4, half of Green's theorem can be demonstrated. The following is a proof of half of the theorem for the simplified area D , a type I region where C 1 and C 3 are curves connected by vertical lines (possibly of zero length).
A parabolic segment is the region bounded by a parabola and line. To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle. The base of this triangle is the given chord of the parabola, and the third vertex is the point on the parabola such that the tangent to the parabola at that point is parallel to the chord.
It also implies that over a closed curve enclosing a region where f(z) is analytic without singularities, the value of the integral is simply zero, or in case the region includes singularities, the residue theorem computes the integral in terms of the singularities. This also implies the path independence of complex line integral for analytic ...
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). [43]
The envelope E 1 is the limit of intersections of nearby curves C t.; The envelope E 2 is a curve tangent to all of the C t.; The envelope E 3 is the boundary of the region filled by the curves C t.
In topology, the Jordan curve theorem (JCT), formulated by Camille Jordan in 1887, asserts that every Jordan curve (a plane simple closed curve) divides the plane into an "interior" region bounded by the curve (not to be confused with the interior of a set) and an "exterior" region containing all of the nearby and far away exterior points.
The Reuleaux triangle is the least symmetric curve of constant width according to two different measures of central asymmetry, the Kovner–Besicovitch measure (ratio of area to the largest centrally symmetric shape enclosed by the curve) and the Estermann measure (ratio of area to the smallest centrally symmetric shape enclosing the curve).
By analyzing the class of curves which lie on such a surface, and the degree to which the surfaces force them to curve in ℝ 3, one can associate to each point of the surface two numbers, called the principal curvatures. Their average is called the mean curvature of the surface, and their product is called the Gaussian curvature.