Let, denote a random sample from a distribution having the pdf f ( x, θ ) for γ < θ < δ.

Let v ∈ T < sub > p </ sub > M be a tangent vector to the manifold at p. Then there is a unique geodesic γ < sub > v </ sub > satisfying γ < sub > v </ sub >( 0 )

Let E → M be a vector bundle with covariant derivative ∇ and γ: I → M a smooth curve parameterized by an open interval I.

A connection ∇ on a vector bundle E → M defines a notion of parallel transport on E along a curve in M. Let γ: → M be a smooth path in M. A section σ of E along γ is said to be parallel if

Let γ be the boundary of B ( z < sub > 0 </ sub >, r ), taken with its positive orientation.

Let γ be a simple rectifiable loop in X around P. The ramification index of ƒ at P is

Let γ: → M be a differentiable curve with γ ( 0 )

Let E be a rank k vector bundle over a smooth manifold M and let ∇ be a connection on E. Given a piecewise smooth loop γ: → M based at x in M, the connection defines a parallel transport map P < sub > γ </ sub >: E < sub > x </ sub > → E < sub > x </ sub >.

Let γ be the small circle of radius ε, Γ the larger, with radius R, then

Let γ ( s ) be a plane curve, parameterized by its arclength s. The unit tangent vector to the curve is, by virtue of the arclength parameterization,

Let γ be a periodic orbit through a point p and S be a local differentiable and transversal section of φ through p, called Poincaré section through p.

Let ( R, M, φ ) be a differentiable dynamical system with periodic orbit γ through p. Let

Let M be a connected Riemannian manifold and p a point of M. A map f defined on a neighborhood of p is said to be a geodesic symmetry, if it fixes the point p and reverses geodesics through that point, i. e. if γ is a geodesic and then It follows that the derivative of the map at p is minus the identity map on the tangent space of p. On a general Riemannian manifold, f need not be isometric, nor can it be extended, in general, from a neighbourhood of p to all of M.

Let γ ( s ) be a regular parametric plane curve, where s is the arc length, or natural parameter.

Let T < sup > a </ sup > be the tangent vector to a given geodesic γ, and X < sup > a </ sup > a vector field along γ connecting it to an infinitesimally near geodesic ( the deviation vector ).

Let M be a pseudo-Riemannian manifold ( or any manifold with an affine connection ) and let p be a point in M. Then for every V in T < sub > p </ sub > M there exists a unique geodesic γ: I → M for which γ ( 0 )

Let the open enemy to it be regarded as a Pandora with her box opened ; ;

Let every policeman and park guard keep his eye on John and Jane Doe, lest one piece of bread be placed undetected and one bird survive.

`` Let him be now ''!!

Let us assume that it would be possible for an enemy to create an aerosol of the causative agent of epidemic typhus ( Rickettsia prowazwki ) over City A and that a large number of cases of typhus fever resulted therefrom.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.

Let Af be the null space of Af.

Let N be a linear operator on the vector space V.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.

Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.

Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.

Let us take a set of circumstances in which I happen to be interested on the legislative side and in which I think every one of us might naturally make such a statement.

Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.

Let this be denoted by Af.

Let it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.

Let not your heart be troubled, neither let it be afraid ''.

The same God who called this world into being when He said: `` Let there be light ''!!

For those who put their trust in Him He still says every day again: `` Let there be light ''!!

Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.

Let her out, let her out -- that would be the solution, wouldn't it??

Let and be differentiable functions, and let D be the total derivative operator.

This result can be formally stated in this manner: Let and be two everywhere differentiable functions.

Let there be a set of differentiable fields φ defined over all space and time ; for example, the temperature T ( x, t ) would be representative of such a field, being a number defined at every place and time.

Let be an interval and be a continuously differentiable function.

Let U be an open set in R < sup > n </ sup > and φ: U → R < sup > n </ sup > an injective differentiable function with continuous partial derivatives, the Jacobian of which is nonzero for every x in U. Then for any real-valued, compactly supported, continuous function f, with support contained in φ ( U ),

Here is Leibniz's argument: Let u ( x ) and v ( x ) be two differentiable functions of x.

Let E and F be two vector bundles over a differentiable manifold M. An R-linear mapping of sections is said to be a kth-order linear differential operator if it factors through the jet bundle J < sup > k </ sup >( E ).

Let M be a differentiable manifold and p a point of M. An affine connection on M allows one to define the notion of a geodesic through the point p.

Let F: U → Y be continuously differentiable and assume that the derivative ( dF )< sub > 0 </ sub >: X → Y of F at 0 is a bounded linear isomorphism of X onto Y.

Let be open ( e. g., an interval ), and consider a differentiable function, with derivative f. The differential df of f, at a point, is defined as a certain linear map of the variable dx.

Let M and N be differentiable manifolds and ƒ: M → N be a differentiable map between them.

Let be a piecewise continuously differentiable function which is periodic with some period.

Let E → M be a smooth vector bundle over a differentiable manifold M. Denote the space of smooth sections of E by Γ ( E ).

Let M be a differentiable manifold and p a point of M. Suppose that f is a function defined in a neighborhood of p, and differentiable at p. If v is a tangent vector to M at p, then the directional derivative of f along v, denoted variously as ( see covariant derivative ), ( see Lie derivative ), or ( see Tangent space # Definition via derivations ), can be defined as follows.

Let f: R < sup > n + m </ sup > → R < sup > m </ sup > be a continuously differentiable function.

: Let f: R < sup > n + m </ sup > → R < sup > m </ sup > be a continuously differentiable function, and let R < sup > n + m </ sup > have coordinates ( x, y ).

Let the mapping be continuously Fréchet differentiable.

Let U be an open set in a manifold M, Ω < sup > 1 </ sup >( U ) be the space of smooth, differentiable 1-forms on U, and F be a submodule of Ω < sup > 1 </ sup >( U ) of rank r, the rank being constant in value over U. The Frobenius theorem states that F is integrable if and only if for every the stalk F < sub > p </ sub > is generated by r exact differential forms.

Let E be a vector bundle of fibre dimension k over a differentiable manifold M. A local frame for E is an ordered basis of local sections of E.

Let X be a differentiable manifold and P a linear differential operator