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Find link is a tool written by Edward Betts.Longer titles found: Integration using Euler's formula (view)

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Euler characteristic
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viewpoint is implicit in Cauchy's proof of Euler's formula given below. There are many proofs of Euler's formula. One was given by Cauchy in 1811, as followsPlanar graph (3,859 words) [view diff] exact match in snippet view article find links to article

Euler's formula, one can then show that these graphs are sparse in the sense that if v ≥ 3: e ≤ 3 v − 6. {\displaystyle e\leq 3v-6.} Euler's formula isJohnson's parabolic formula (1,004 words) [view diff] exact match in snippet view article find links to article

can fail by compression. One way to calculate buckling is to utilize Euler's formula, which produces a critical stress vs. slenderness curve such as theJohann F. C. Hessel (1,140 words) [view diff] exact match in snippet view article find links to article

specific examples of compound crystals (aka double crystals) for which Euler's formula for convex polyhedra failed. In this case, the sum of the valence (degree)Simon Antoine Jean L'Huilier (290 words) [view diff] exact match in snippet view article find links to article

mathematical analysis and topology, and in particular the generalization of Euler's formula for planar graphs. He won the mathematics section prize of the BerlinPolyhedral combinatorics (2,189 words) [view diff] exact match in snippet view article find links to article

important relation among the coefficients of the ƒ-vector of a polytope is Euler's formula Σ(−1)ifi = 0, where the terms of the sum range over the coefficientsLewis' law (575 words) [view diff] exact match in snippet view article find links to article

≈ 6 {\textstyle {\bar {n}}\approx 6} , which can be traced back to Euler's formula for polygons. Frederic Thomas Lewis noticed that epidermal cells displayEuler's Gem (887 words) [view diff] exact match in snippet view article find links to article

version of the Gauss–Bonnet theorem (later seen to be equivalent to Euler's formula). It surveys the life of Euler, his discovery in the early 1750s thatEuler's continued fraction formula (3,211 words) [view diff] no match in snippet view article find links to article

In the analytic theory of continued fractions, Euler's continued fraction formula is an identity connecting a certain very general infinite series withPolyhedron (7,320 words) [view diff] no match in snippet view article find links to article

In geometry, a polyhedron (plural polyhedra or polyhedrons) is a three dimensional shape with flat polygonal faces, straight edges and sharp corners orSine and cosine transforms (1,012 words) [view diff] case mismatch in snippet view article find links to article

= ∫ − ∞ ∞ f ( t ) ( cos ( 2 π ν t ) − i sin ( 2 π ν t ) ) d t Euler's Formula = ( ∫ − ∞ ∞ f ( t ) cos ( 2 π ν t ) d t ) − i ( ∫ − ∞ ∞ f ( t ) sinCuboid (570 words) [view diff] exact match in snippet view article find links to article

hexahedron, right rectangular prism, or rectangular parallelepiped. By Euler's formula the numbers of faces F, of vertices V, and of edges E of any convexLazy caterer's sequence (701 words) [view diff] exact match in snippet view article find links to article

{n(n+1)}{2}}={\frac {n^{2}+n+2}{2}}.} Floyd's triangle Moore, T. L. (1991), "Using Euler's formula to solve plane separation problems", The College Mathematics JournalViète's formula (1,594 words) [view diff] exact match in snippet view article find links to article

Another derivation is possible based on trigonometric identities and Euler's formula. By repeatedly applying the double-angle formula sin x = 2 sin Dual graph (6,487 words) [view diff] exact match in snippet view article find links to article

constructed as the adhesion of a tetrahedron with its dual. It follows from Euler's formula that every self-dual graph with n vertices has exactly 2n − 2 edgesEulerian poset (397 words) [view diff] exact match in snippet view article find links to article

itself, is an Eulerian lattice. The odd–even condition follows from Euler's formula. Any simplicial generalized homology sphere is an Eulerian latticeKepler–Poinsot polyhedron (2,131 words) [view diff] exact match in snippet view article find links to article

are no longer part of the polyhedral surface, and can disappear. Now Euler's formula holds: 60 − 90 + 32 = 2. However, this polyhedron is no longer theEuler Book Prize (612 words) [view diff] exact match in snippet view article find links to article

(Princeton University Press, 2008). Richeson relates the history of Euler's formula V − E + F = 2 connecting the numbers of vertices, edges, and facesSimplicial sphere (507 words) [view diff] exact match in snippet view article find links to article

polytope in the Euclidean space is a simplicial d-sphere. It follows from Euler's formula that any simplicial 2-sphere with n vertices has 3n − 6 edges and 2nSpiral of Theodorus (1,073 words) [view diff] exact match in snippet view article find links to article

was proposed and answered in (Davis 2001, pp. 37–38) by analogy with Euler's formula for the gamma function as an interpolant for the factorial functionStacked polytope (422 words) [view diff] exact match in snippet view article find links to article

two-dimensional faces are determined from the number of vertices by Euler's formula, regardless of whether the polyhedron is stacked, but this is not trueKuratowski's theorem (943 words) [view diff] exact match in snippet view article find links to article

as may be shown either by a case analysis or an argument involving Euler's formula. Additionally, subdividing a graph cannot turn a nonplanar graph intoFáry's theorem (1,222 words) [view diff] exact match in snippet view article find links to article

and c are the only vertices in G. Thus, we may assume that n ≥ 4. By Euler's formula for planar graphs, G has 3n − 6 edges; equivalently, if one definesGrinberg's theorem (854 words) [view diff] exact match in snippet view article find links to article

_{k\geq 3}(k-2)(f_{k}-g_{k})=0.} The proof is an easy consequence of Euler's formula. As a corollary of this theorem, if an embedded planar graph has onlyMünchenstein rail disaster (723 words) [view diff] exact match in snippet view article find links to article

occurred in Europe. His investigation of the collapse revealed that Euler's formula for buckling, which had hitherto been used to calculate design loadsJacket matrix (843 words) [view diff] case mismatch in snippet view article find links to article

4 T = I 4 . {\displaystyle \ J_{4}^{T}J_{4}=J_{4}J_{4}^{T}=I_{4}.} Euler's Formula: e i π + 1 = 0 {\displaystyle e^{i\pi }+1=0} , e i π = cos π + iDischarging method (discrete mathematics) (971 words) [view diff] case mismatch in snippet view article

sum of all the face lengths equals twice the number of edges. Using Euler's Formula, it's easy to see that the sum of all the charges is 12: ∑ f ∈ F 6Crossing number inequality (1,225 words) [view diff] exact match in snippet view article find links to article

n vertices with no crossings, and is thus a planar graph. But from Euler's formula we must then have e − cr(G) ≤ 3n, and the claim follows. (In fact wePolytope (2,942 words) [view diff] exact match in snippet view article find links to article

n_{j}} is the number of j {\displaystyle j} -faces. This generalizes Euler's formula for polyhedra. The Gram–Euler theorem similarly generalizes the alternatingMac Lane's planarity criterion (1,344 words) [view diff] exact match in snippet view article find links to article

one and the induction follows. Alternatively, it is possible to use Euler's formula to show that the number of cycles in this collection equals the circuitMac Lane's planarity criterion (1,344 words) [view diff] exact match in snippet view article find links to article

one and the induction follows. Alternatively, it is possible to use Euler's formula to show that the number of cycles in this collection equals the circuitThomas Kirkman (1,274 words) [view diff] exact match in snippet view article find links to article

enumeration problems concerning polyhedra, beginning with a proof of Euler's formula and concentrating on simple polyhedra (the polyhedra in which eachJurij Vega (1,310 words) [view diff] exact match in snippet view article find links to article

5}\right)-\arctan \left({1 \over 239}\right)} with his formula, which is equal to Euler's formula from 1755: π 4 = 5 arctan ( 1 7 ) + 2 arctan ( 3 79 ) , {\displaystyleMurderous Maths (2,033 words) [view diff] exact match in snippet view article find links to article

origami, circles: chord; tangent; angle theorems, regular solids, Euler's formula, ellipses, Geometric proof of Pythagoras' Theorem.) The Key To TheAutomedian triangle (1,457 words) [view diff] exact match in snippet view article find links to article

could not be used to form the sides of a triangle. Consequently, using Euler's formula that generates primitive Pythagorean triangles it is possible to generateOccurrences of Grandi's series (1,690 words) [view diff] exact match in snippet view article find links to article

edge, one face, and generally exactly one cell of every dimension, Euler's formula V − E + F − · · · for the Euler characteristic of S returns 1 − 1 +Ludwig Schläfli (1,905 words) [view diff] exact match in snippet view article find links to article

theory and finds, among other things, the higher-dimensional version of Euler's formula. He determines the regular polytopes, i.e. the n {\displaystyle n}Francesco Maurolico (1,576 words) [view diff] exact match in snippet view article find links to article

Compaginationes solidorum regularium (1537) includes a statement of Euler's formula V − E + F = 2 {\displaystyle V-E+F=2} for the Platonic solids, longGraph theory (5,901 words) [view diff] exact match in snippet view article find links to article

problem, carried on with the analysis situs initiated by Leibniz. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedronHidden-line removal (1,401 words) [view diff] exact match in snippet view article find links to article

sphere and with faces topologically equivalent to disks, according to Euler's formula, there are Θ(n) faces. Testing Θ(n2) line segments against Θ(n) facesEuler's critical load (1,395 words) [view diff] exact match in snippet view article find links to article

should be used. The following assumptions are made while deriving Euler's formula: The material of the column is homogeneous and isotropic. The compressiveJacobi symbol (2,118 words) [view diff] exact match in snippet view article find links to article

random number a, calculate the Jacobi symbol (a/n) and compare it with Euler's formula; if they differ modulo n, then n is composite; if they have the samePlatonic solid (5,095 words) [view diff] exact match in snippet view article find links to article

pF=2E=qV.\,} The other relationship between these values is given by Euler's formula: V − E + F = 2. {\displaystyle V-E+F=2.\,} This can be proved in manyImre Lakatos (4,887 words) [view diff] exact match in snippet view article find links to article

extensive footnotes. Lakatos termed the polyhedral counterexamples to Euler's formula monsters and distinguished three ways of handling these objects: FirstlyHeegner number (3,299 words) [view diff] exact match in snippet view article find links to article

n = 1, ..., 40, is related to the Heegner number 163 = 4 · 41 − 1. Euler's formula, with n {\displaystyle n} taking the values 1,... 40 is equivalentEuler's totient function (6,084 words) [view diff] exact match in snippet view article find links to article

some element of Cn, the formula follows. In the article Root of unity Euler's formula is derived by using this argument in the special case of the multiplicativeFour color theorem (5,616 words) [view diff] exact match in snippet view article find links to article

is shared by two regions, we have that 2e = 3f. This together with Euler's formula, v − e + f = 2, can be used to show that 6v − 2e = 12. Now, the degreeFour color theorem (5,616 words) [view diff] exact match in snippet view article find links to article

is shared by two regions, we have that 2e = 3f. This together with Euler's formula, v − e + f = 2, can be used to show that 6v − 2e = 12. Now, the degreeDessin d'enfant (3,836 words) [view diff] exact match in snippet view article find links to article

trees. Any embedding of a tree has a single region, and therefore by Euler's formula lies in a spherical surface. The corresponding Belyi pair forms a transformationAugustin-Louis Cauchy (5,465 words) [view diff] exact match in snippet view article find links to article

three given circles—which he discovered in 1805, his generalization of Euler's formula on polyhedra in 1811, and in several other elegant problems. More importantSpin (physics) (10,195 words) [view diff] exact match in snippet view article

all n-fold tensor products of Pauli matrices. The analog formula of Euler's formula in terms of the Pauli matrices: e i θ ( n ^ ⋅ σ ) = I cos ( θ ) +Incircle and excircles of a triangle (5,554 words) [view diff] no match in snippet view article find links to article

Mifflin, Boston, 1929: p. 187. Emelyanov, Lev, and Emelyanova, Tatiana. "Euler’s formula and Poncelet’s porism", Forum Geometricorum 1, 2001: pp. 137–140. Altshiller-CourtScientific method (16,984 words) [view diff] exact match in snippet view article find links to article

1991 p. 100 See the development, by generations of mathematicians, of Euler's formula for polyhedra as documented by Lakatos, Imre (1976), Proofs and refutationsRiemann zeta function (9,244 words) [view diff] exact match in snippet view article find links to article

arithmetic. Since the harmonic series, obtained when s = 1, diverges, Euler's formula (which becomes ∏p p/p − 1) implies that there are infinitely many primesCycle basis (3,270 words) [view diff] exact match in snippet view article find links to article

the vertices of the graph) and the remaining faces are bounded. By Euler's formula for planar graphs, there are exactly m − n + 1 {\displaystyle m-n+1}Small cancellation theory (4,205 words) [view diff] exact match in snippet view article find links to article

roughly - the average excess of vertices + faces - edges (which, by Euler's formula, must total 2) and, by showing, in a particular group, that this isBernoulli number (11,892 words) [view diff] exact match in snippet view article find links to article

second, the numerators of the first column are the denominators of Euler's formula. The first column is −1/2 × OEIS: A163982. The sequence Sn has anotherGeneralized continued fraction (8,676 words) [view diff] exact match in snippet view article find links to article

{a_{0}x}{a_{1}+x-}}{\frac {a_{1}x}{a_{2}+x-}}\cdots {\frac {a_{n-1}x}{a_{n}+x}}.\,} Euler's formula connecting continued fractions and series is the motivation for theQuadratic reciprocity (9,211 words) [view diff] exact match in snippet view article find links to article

section are true for Jacobi symbols as long as the symbols are defined. Euler's formula may be written ( a m ) = ( a m ± 4 a n ) , n ∈ Z , m ± 4 a n > 0. {\displaystylePlanar separator theorem (8,687 words) [view diff] exact match in snippet view article find links to article

G with no vertex having degree greater than 3. From a corollary of Euler's formula, the number of vertices in the resulting graph will be n ≤ 6n0 -12