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The curvature and torsion functions along a space curve determine it up to an orientation-preserving isometry.

Consider the forms Q for which the generic characters chi_i(Q) are equal to some preassigned array of signs e_i=1 or -1, e_1,e_2,...,e_r, subject to product_(i=1)^(r)e_i=1. ...

In the most commonly used convention (e.g., Apostol 1967, pp. 205-207), the second fundamental theorem of calculus, also termed "the fundamental theorem, part II" (e.g., ...

For a simple continued fraction x=[a_0,a_1,...] with convergents p_n/q_n, the fundamental recurrence relation is given by p_nq_(n-1)-p_(n-1)q_n=(-1)^(n+1).

Two unit-speed plane curves which have the same curvature differ only by a Euclidean motion.

If M is continuous and int_a^bM(x)h(x)dx=0 for all infinitely differentiable h(x), then M(x)=0 on the open interval (a,b).

If two single-valued continuous functions kappa(s) (curvature) and tau(s) (torsion) are given for s>0, then there exists exactly one space curve, determined except for ...

The abscissas of the N-point Gaussian quadrature formula are precisely the roots of the orthogonal polynomial for the same interval and weighting function.

On a Riemannian manifold, there is a unique connection which is torsion-free and compatible with the metric. This connection is called the Levi-Civita connection.

Let F_0 and F_1 denote two directly similar figures in the plane, where P_1 in F_1 corresponds to P_0 in F_0 under the given similarity. Let r in (0,1), and define ...