Sentence examples for plane section from inspiring English sources

end{aligned} (2.11) Furthermore, the scalar curvature ρ for a submanifold M of an almost contact manifold M̃ is given by begin{aligned} rho=sum_{1leq ineq jleq n}K e_{i} wedge e_{j}), end{aligned} (2.12) where (K e_{i}wedge e_{j})) is the sectional curvature of plane section spanned by (e_{i}) and (e_{j}).

where K ( e i ∧ e j ) is the sectional curvature of the plane section spanned by e i and e j at x ∈ M. Let Π k be a k-plane section of T x M, and let { e 1, …, e k } be any orthonormal basis of  Π k.

where K ( e i ∧ e j ) and K ¯ ( e i ∧ e j ) denote the sectional curvature of the plane section spanned by e i and e j at x in the submanifold M and in the ambient manifold M ¯, respectively.

Then the mean curvature vector H is given by H = ∑ r = n + 1 n + p ( 1 n ∑ i = 1 n h i i r ) e r, and the squared norm of h over dimension n is denoted by C and is called the Casorati curvature of the submanifold M. Therefore we have C = 1 n ∑ r = n + 1 n + p ∑ i, j = 1 n ( h i j r ) 2. Let K ( e i ∧ e j ), 1 ≤ i < j ≤ n, denote the sectional curvature of the plane section spanned by e i and e j.

Then we denote by g the metric tensor induced on M. Let (K pi)) be the sectional curvature of M associated with a plane section (pisubset T_{p}M), (pin M).

We denote by K the sectional curvature of M associated with a plane section π ⊂ T p M, p ∈ M. If { e 1, …, e n } is an orthonormal basis of the tangent space T p M, the scalar curvature τ at p is defined by τ ( p ) = ∑ 1 ≤ i < j ≤ n K ( e i ∧ e j ).

Conic sections can be regarded as plane sections of a right circular cone (see the figure).

These beam elements are formulated from classical beam theory, with a basic assumption that "plane sections remain plane" during bending.

A new analytical (non-finite element) model for simulation of tubular hydroforming plane sections has been introduced.

This phenomenon called torsion induced vertical slip is an important issue, which would make the assumption plane sections remain plane invalid.

Homology metrics have been used to assess the connectivity of grain boundary networks in plane sections of polycrystals.