Word Finder - theorem

definition

A mathematical statement of some importance that has been proven to be true. Minor theorems are often called propositions. Theorems which are not very interesting in themselves but are an essential part of a bigger theorem's proof are called lemmas.

definition

A mathematical statement that is expected to be true

definition

A syntactically correct expression that is deducible from the given axioms of a deductive system.

definition

To formulate into a theorem.

This is the true standpoint from which the theorem should be regarded.

By means of this theorem it can be shown that, whatever the value of n may be, f 1 + (plq)(i)x+(p/q)(2)x2+...

This enabled David Hilbert to produce a very simple unsymbolic proof of the same theorem.

The binomial theorem gives a formula for writing down the coefficient of any stated term in the expansion of any stated power of a given binomial.

The argument involves the theorem that, if 0 is a positive quantity less than I, 0 t can be made as small as we please by taking t large enough; this follows from the fact that tlog 0 can be made as large (numerically) as we please.

This is commonly called Stokes's theorem.

Legendre, in 1783, extended Maclaurin's theorem concerning ellipsoids of revolution to the case of any spheroid of revolution where the attracted point, instead of being limited to the axis or equator, occupied any position in space; and Laplace, in his treatise Theorie du mouvement et de la figure elliptique des planetes (published in 1784), effected a still further generalization by proving, what had been suspected by Legendre, that the theorem was equally true for any confocal ellipsoids.

The first formal proof of Lagrange's theorem for the development in a series of an implicit function was furnished by Laplace, who gave to it an extended generality.

If we represent this expression by f (x), the expression obtained by changing x into x-+-h is f(x+h); and each term of this may be expanded by the binomial theorem.

The relation, when written in the form (23), is known as Vandermonde's theorem.

It is scarcely necessary to remark that in all such cases the calculation applies in the first instance to homogeneous light, and that, in accordance with Fourier's theorem, each homogeneous component of a mixture may be treated separately.

Thus, if a= d, we should have 1=2+4+77(49425+...) which is true by a known theorem.

A full discussion would call for the formal application of Fourier's theorem, but some conclusions of importance are almost obvious.

This theorem was published in 1643, at the end of his treatise De motu gravium projectorum, and it was confirmed by the experiments of Raffaello Magiotti on the quantities of water discharged from different ajutages under different pressures (1648).

The theorem of Torricelli was employed by many succeeding writers, but particularly by Edme Mariotte (1620-1684), whose Traite du mouvement des eaux, published after his death in the year 1686, is founded on a great variety of well-conducted experiments on the motion of fluids, performed at Versailles and Chantilly.

This theorem is called generally the principle of Archimedes.

As the molten metal is run in, the upward thrust on the outside mould, when the level has reached PP', is the weight of metal in the volume generated by the revolution of APQ; and this, by a theorem of Archimedes, has the same volume as the cone ORR', or rya, where y is the depth of metal, the horizontal sections being equal so long as y is less than the radius of the outside FIG.

In particular, for a jet issuing into the atmosphere, where p=P, q 2 /2g = h - z, (9) or the velocity of the jet is due to the head k-z of the still free surface above the orifice; this is Torricelli's theorem (1643), the foundation of the science of hydrodynamics.

The circulation being always zero round a small plane curve passing through the axis of spin in vortical motion, it follows conversely that a vortex filament is composed always of the same fluid particles; and since the circulation round a cross-section of a vortex filament is constant, not changing with the time, it follows from the previous kinematical theorem that aw is constant for all time, and the same for every cross-section of the vortex filament.

The binomial theorem is a celebrated theorem, originally due to Sir Isaac Newton, by which any power of a binomial can be expressed as a series.

In its modern form the theorem, which is true for all values of n, is written as (x +a) n -1+ I.

It is a fundamental theorem in attractions that a thin spherical shell of matter which attracts according to the potential law of the inverse square acts on all external points as of a if it were concentrated at its centre.

Stokes's theorem becomes an obvious truism if applied to an incompressible fluid.

Let us apply the above theorem to the case of a small parallelepipedon or rectangular prism having sides dx, dy, dz respectively, its centre having co-ordinates (x, y, z).

Hence the total flux is - (+ d2V d 2 V d2V dye + dz2) dy dz, dx2 and by the previous theorem this must be equal to 4'rrp dxdydz.

Hence if dS and dS' are the areas of the ends, and +E and - E' the oppositely directed electric forces at the ends of the tube, the surface integral of normal force on the flux over the tube is EdS - E'dS' (20), and this by the theorem already given is equal to zero, since the tube includes no electricity.

We begin with a general dynamical theorem, whose special application, when the dynamical system is identified with a gas, will appear later.

But Landen's capital discovery is that of the theorem known by his name (obtained in its complete form in the memoir of 1775, and reproduced in the first volume of the Mathematical Memoirs) for the expression of the arc of an hyperbola in terms of two elliptic arcs.

The formula applied can then be either Simpson's rule or a rule based on Gauss's theorem for two ordinates (§ 56).

Fourier's theorem asserts that such a curve may be built up by the superposition, or addition of ordinates, of a series of sine curves of wave-lengths X, IX, 3A, 4A...

Fourier's theorem can also be usefully applied to the disturbance of a source of sound under certain conditions.

Now we may resolve these trains by Fourier's theorem into harmonics of wave-lengths X, 2X, 3A, &c., where X=2AB and the conditions as to the values of y can be shown to require that the harmonics shall all have nodes, coinciding with the nodes of the fundamental curve.

If then we resolve Ahbkc into harmonics by Fourier's theorem, we may follow the motion of the separate harmonics, and their superposition will give the form of the string at any instant.

We see, then, that the conditions for the application of Fourier's theorem are equivalent to saying that all disturbances will travel along the system with the same velocity.

In many vibrating systems this does not hold, and then Fourier's theorem is no longer an appropriate resolution.

But with the sole exception of proving that the volumes of spheres are in the triplicate ratio of their diameters, a theorem probably due to Eudoxus, no mention is made of its mensuration.

To Legendre is due the theorem known as the law of quadratic reciprocity, the most important general result in the science of numbers which has been discovered since the time of P. de Fermat, and which was called by Gauss the " gem of arithmetic."

Legendre shows that Maclaurin's theorem with respect to confocal ellipsoids is true for any position of the external point when the ellipsoids are solids of revolution.

The third memoir relates to Laplace's theorem respecting confocal ellipsoids.

The best known of these, which is called Legendre's theorem, is usually given in treatises on spherical trigonometry; by means of it a small spherical triangle may be treated as a plane triangle, certain corrections being applied to the angles.

Legendre's theorem is a fundamental one in geodesy, and his contributions to the subject are of the greatest importance.

A remarkable theorem is I -x.

The theorem for angle-bisection which Vieta used was not that of Archimedes, but that which would now appear in the form I - cos 0 = 2 sin e 20.

With the mention of the Porisms of Euclid we have an account of the relation of porism to theorem and problem.

To the former belong the theorems (t), (2), and (3), and to the latter especially the theorem (4), and also, probably, his solution of the two practical problems. We infer, then, [t] that Thales must have known the theorem that the sum of the three angles of a triangle are equal to two right angles.

No doubt we are informed by Proclus, on the authority of Eudemus, that the theorem Euclid i.

The theorem, then, seems to have been arrived at by induction, and may have been suggested by the contemplation of floors or walls covered with tiles of the form of equilateral triangles, or squares, or hexagons.

The knowledge of this theorem is distinctly attributed to Thales by Plutarch, and it was probably made use of also in his determination of the distance of a ship at sea.

With Locke, Hume professes to regard this problem as virtually covered or answered by the fundamental psychological theorem; but the superior clearness of his reply enables us to mark with perfect precision the nature of the difficulty inherent in the attempt to regard the two as identical.

In this way he established the famous theorem that the intersections of the three pairs of opposite sides of a hexagon inscribed in a conic are collinear.