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A living hinge or integral hinge is a thin flexible hinge (flexure bearing).It is made from the same material as the two rigid pieces it connects. It is typically thinned or cut to allow the rigid pieces to bend along the line of the hinge.
In geotechnical engineering, a discontinuity (often referred to as a joint) is a plane or surface that marks a change in physical or chemical characteristics in a soil or rock mass. A discontinuity can be, for example, a bedding , schistosity , foliation , joint , cleavage , fracture , fissure , crack, or fault plane.
Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the ...
(Note that the term − f (x) / 2 ( 1 / b − x − 1 / a − x ) in the second equivalent definition above is missing in Ang but this is corrected in the errata sheet of the book.) Integral equations containing Hadamard finite part integrals (with f (t) unknown) are termed hypersingular integral equations. Hypersingular ...
For example, if the domain of integration is defined as the plane region between two -coordinates and the graphs of two functions, it will often happen that the domain has corners. In such a case, the corner points mean that Ω {\displaystyle \Omega } is not a smooth manifold with boundary, and so the statement of Stokes' theorem given above ...
The semimartingales form the largest class of processes for which the Itō integral can be defined. Linear combinations of semimartingales are semimartingales. Products of semimartingales are semimartingales, which is a consequence of the integration by parts formula for the Itō integral. The quadratic variation exists for every semimartingale.
The Riemann–Stieltjes integral admits integration by parts in the form () = () () ()and the existence of either integral implies the existence of the other. [2]On the other hand, a classical result [3] shows that the integral is well-defined if f is α-Hölder continuous and g is β-Hölder continuous with α + β > 1 .
It equates the surface integral of the curl of a vector field to the above line integral taken around the boundary of the surface. Another way one can define the curl vector of a function F at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosing p divided by the volume enclosed, as the shell ...