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Page "Laplace–Runge–Lenz vector" ¶ 28

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Let and Cartesian

Let x, y, z be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors.

" Let X be the unit Cartesian square ×, and let ~ be the equivalence relation on X defined by ∀ a, b ∈ (( a, 0 ) ~ ( a, 1 ) ∧ ( 0, b ) ~ ( 1, b )).

Let M be a smooth manifold of dimension n ; for instance a surface ( in the case n = 2 ) or hypersurface in the Cartesian space R n + 1 .

Let M × M be the Cartesian product of M with itself.

Suppose that c is a simple closed curve in a closed, orientable surface S. Let A be a tubular neighborhood of c. Then A is an annulus and so is homeomorphic to the Cartesian product of

Doolittle has mistakenly taught the bomb Cartesian doubt, the bomb determines itself to be God, states " Let there be light ," and promptly detonates.

Let and coordinates

Let ρ, θ, and φ be spherical coordinates for the source point P. Here θ denotes the angle with the vertical axis, which is contrary to the usual American mathematical notation, but agrees with standard European and physical practice.

Let us now imagine a Lorentz transformation to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector.

Let the action be invariant under certain transformations of the space – time coordinates x μ and the fields φ

Let x: y: z be a variable point in trilinear coordinates, and let u

Let be the coordinates of a rotation by α around the axis as previously described.

Let A, B, C denote the vertex angles of the reference triangle, and let x: y: z be a variable point in trilinear coordinates ; then an equation for the Euler line is

Let k be an algebraically closed field and let P n be a projective n-space over k. Let f ∈ k ..., x n be a homogeneous polynomial of degree d. It is not well-defined to evaluate f on points in P n in homogeneous coordinates.

Let X, Y, Z be the ambient coordinates in R 3 .

Let P = ( r, θ ) be a point on a given curve defined by polar coordinates and let O denote the origin.

Let Λ be the lattice in R d consisting of points with integer coordinates ; Λ is the character group, or Pontryagin dual, of R d .

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

Let denote the linear operator which sets all coordinates, to zero.

Let be the internal coordinates of a point of.

Let the local coordinates be called.

Let M be a differentiable manifold, of dimension n, and v a vector field on M. Suppose that x is an isolated zero of v, and fix some local coordinates near x.

Let be a different coordinate system and let be the associated change of coordinates diffeomorphism of Euclidean space to itself.

Let w k be the coefficients of the vector field v in the y coordinates.

Let denote the sum of these points,, then its coordinates are given by:

Let and be

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??

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