9807

Slanted Line Integral

Generally speaking, the line integral of a function over a given curve is defined by first selecting points on the curve, then evaluating the function's values at these points, and afterterwards, refer to the limit of the evaluated expression as the number of selected points approaches infinity.
The definition of the slanted line integral is of a different nature - it uses no division of the given curve, and the calculation is performed mostly in a domain which is not on the given curve itself. Let f:{R}2{R} be an integrable function and let F:R2R be its comulative distribution function, . Let AABB be a given continuous and tendable curve (i.e., its tendency is defined for all its interior points). Suppose that AB is uniformly tended (i.e., its tendency is constant for all its interior points). Then the slanted line integral of F over AB is defined as the following summation:
,
where D is the positive domain of the curve (i.e., the domain whose boundary are AB and two lines which are paralel to the axes, such that D is on the left hand-side of AB), τ(AB) is the tendency of the uniformly tended curve AB, and τ(A), τ(B) are the tendencies of the curve at the points A,B respectively. The name "Slanted Integral" is due to the fact that this integration method is effected by the rotation angle of the coordinates system, since the domain D and the calculation of the comulative distribution function are both dependent of that angle. Hence, it is assumed that the coordinates system is static and given in advance. In this demonstration the axes are paralel to the edges of the figure.
In this demonstration, the user may manipulate the locations of the points A,A,B,B, the orientation of the curve, and the curvature of the sub-curve AB , to watch how these changes effect each of the parameters in the definition of the slanted line integral above: the domain D, the curve's tendency τ(AB), and the pointwise tendencies, τ(A) and τ(B).
  • Contributed by: Amir Finkelstein


Thumbnail: The pointwise tendencies at A,B are both zero, and the curve's tendency is -1. Hence, the definition of the slanted line integral of F over AB results, in this case, with .
Snapshot 1: Tendency of a curve at a given point is defined if and only if the curve is defined in the right and left neighborhoods of the point. Hence, in this case the pointwise tendencies at A,B are both undefined, and so the slanted line integral of F over AB is undefined as well.
Snapshot 2: In case the tendency of the curve AB is zero, then there is no domain D to integrate over, and no point C. Here the pointwise tendencies are τ(A)-1,τ(B)+1, hence the definition of the slanted line integral of F over AB results, in this case, with .
Snapshot 3: The curvature of the curve AB effects only the domain D. Note that the domain's position with respect to the curve is changed once the orientation is reverted - because the left hand-side of the curve also reverts. Note that the lines AA' and BB' may intersect this domain. Here the pointwise tendencies are τ(A)τ(B)=-1, and the curve's tendency is -1. Hence, the definition of the slanted line integral of F over AB results, in this case, with .
Slanted line integral is an integration method that extends the integration method found in the discrete Green's theorem to more general types of domains - i.e., domains which are not necessarily formed by a finite unification of rectangles. In turn, this semi-discrete integration method results with a theorem which extends the discrete Green's theorem. More details are available in a preprint. More details regarding the definition of tendency are also available in a previously submited demonstration.
 
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