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## Slanted Line IntegralThumbnail: 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. "Slanted Line Integral" from The Wolfram Demonstrations Project http://demonstrations.wolfram.com/SlantedLineIntegral/ Contributed by: Amir Finkelstein |