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The curve () describes the deflection of the beam in the direction at some position (recall that the beam is modeled as a one-dimensional object). q {\displaystyle q} is a distributed load, in other words a force per unit length (analogous to pressure being a force per area); it may be a function of x {\displaystyle x} , w {\displaystyle w ...
The property of remaining a constant length under load has been made use of in length metrology. When metal bars were developed as physical standards for length measures, they were calibrated as marks made on a length measured along the neutral plane. This avoided the minuscule changes in length, owing to the bar sagging under its own weight.
Thus the length of a curve is a non-negative real number. Usually no curves are considered which are partly spacelike and partly timelike. In theory of relativity, arc length of timelike curves (world lines) is the proper time elapsed along the world line, and arc length of a spacelike curve the proper distance along the curve.
Since a convex curve intersects almost every line either twice or not at all, the unoriented Crofton formula for convex curves can be stated without numerical factors: the measure of the set of straight lines which intersect a convex curve is equal to its length. The same formula (with the same multiplicative constants) apply for hyperbolic ...
The coordinate-independent definition of the square of the line element ds in an n-dimensional Riemannian or Pseudo Riemannian manifold (in physics usually a Lorentzian manifold) is the "square of the length" of an infinitesimal displacement [2] (in pseudo Riemannian manifolds possibly negative) whose square root should be used for computing curve length: = = (,) where g is the metric tensor ...
Thus, it enables one to calculate the lengths of curves on the surface and the areas of regions on the surface. The line element ds may be expressed in terms of the coefficients of the first fundamental form as d s 2 = E d u 2 + 2 F d u d v + G d v 2 . {\displaystyle ds^{2}=E\,du^{2}+2F\,du\,dv+G\,dv^{2}\,.}
The cantilever method is an approximate method for calculating shear forces and moments developed in beams and columns of a frame or structure due to lateral loads. The applied lateral loads typically include wind loads and earthquake loads, which must be taken into consideration while designing buildings.
For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of the curve projected onto the surface's tangent plane. More generally, in a given manifold M ¯ {\displaystyle {\bar {M}}} , the geodesic curvature is just the usual curvature of γ {\displaystyle \gamma } (see below).