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Longer titles found: Integration using Euler's formula (view)

searching for Euler's formula 61 found (366 total)

Euler characteristic (3,461 words) [view diff] exact match in snippet view article find links to article

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 follows

Pick's theorem (2,339 words) [view diff] exact match in snippet view article find links to article

way) as the basis for a proof of Euler's formula. Alternative proofs of Pick's theorem that do not use Euler's formula include the following. One can recursively

Planar graph (4,541 words) [view diff] exact match in snippet view article find links to article

characterize the planar graphs via a system of equations modulo 2. Euler's formula states that if a finite, connected, planar graph is drawn in the plane

Johann F. C. Hessel (1,137 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)

Johnson's parabolic formula (1,075 words) [view diff] exact match in snippet view article find links to article

yield stress of the material to the critical buckling stress given by Euler's formula relating the slenderness ratio to the stress required to buckle a column

Simon Antoine Jean L'Huilier (398 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 Berlin

Lewis' law (568 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 display

Euler's Gem (934 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 that

Polyhedral combinatorics (2,304 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 coefficients

Dual graph (6,607 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 edges

Eulerian poset (407 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 lattice

Small stellated dodecahedron (768 words) [view diff] exact match in snippet view article find links to article

meeting at 30 edges and 12 vertices, we can compute its genus using Euler's formula V − E + F = 2 − 2 g {\displaystyle V-E+F=2-2g} and conclude that the

Integer lattice (516 words) [view diff] exact match in snippet view article find links to article

Aigner, Martin; Ziegler, Günter M. (2018). "Three applications of Euler's formula: Pick's theorem". Proofs from THE BOOK (6th ed.). Springer. pp. 93–94

Spiral of Theodorus (1,156 words) [view diff] exact match in snippet view article find links to article

was proposed and answered by Philip J. Davis in 2001 by analogy with Euler's formula for the gamma function as an interpolant for the factorial function

Lazy caterer's sequence (823 words) [view diff] exact match in snippet view article find links to article

Vol. 1. New York: Dover Publications. Moore, T. L. (1991), "Using Euler's formula to solve plane separation problems", The College Mathematics Journal

Kepler–Poinsot polyhedron (2,312 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 the

Simplicial sphere (515 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 2n

Viète's formula (2,271 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. Repeatedly applying the double-angle formula sin ⁡ x = 2 sin ⁡ x 2

Euler Book Prize (724 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 faces

Triangular prism (1,608 words) [view diff] case mismatch in snippet view article find links to article

Wrenn; Williams, Gordon (2009). "Exploring Polyhedra and Discovering Euler's Formula". In Hopkin, Brian (ed.). Resources for Teaching Discrete Mathematics:

Stacked polytope (423 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 true

1105 (number) (667 words) [view diff] exact match in snippet view article

Encyclopedia of Integer Sequences. OEIS Foundation. Gould, H. W. (1978). "Euler's formula for n {\displaystyle n} th differences of powers". The American Mathematical

Kuratowski's theorem (1,074 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 into

Fáry's theorem (1,261 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 defines

Francesco Maurolico (1,703 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, long

Thomas Kirkman (1,275 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 each

Münchenstein rail disaster (731 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 loads

Polytope (3,119 words) [view diff] exact match in snippet view article find links to article

number of j {\displaystyle j} -dimensional faces. This generalizes Euler's formula for polyhedra. The Gram–Euler theorem similarly generalizes the alternating

Grinberg's theorem (1,161 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 only

Cluster decomposition (1,309 words) [view diff] exact match in snippet view article find links to article

internal momenta, leaving V-(I-L) delta functions unfixed. A form of Euler's formula states that any graph with C disjoint connected components satisfies

Crossing number inequality (1,392 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 we

Murderous Maths (1,865 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 The

Invariant decomposition (2,082 words) [view diff] exact match in snippet view article find links to article

each F i {\displaystyle F_{i}} squares to a scalar and thus follows Euler's formula: R i = e F i = cosh ( λ i ) + sinh ( λ i ) λ i F i . {\displaystyle

Discharging method (discrete mathematics) (1,075 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 6

Mac Lane's planarity criterion (1,365 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 circuit

Hidden-line removal (1,403 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) faces

The Housekeeper and the Professor (1,486 words) [view diff] exact match in snippet view article find links to article

money from the Professor. However, after the Professor writes down Euler's formula during this confrontation, the Widow immediately comes to accept the

Automedian triangle (1,750 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 generate

Occurrences of Grandi's series (1,721 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 +

Graph theory (6,237 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 polyhedron

Jacobi symbol (2,390 words) [view diff] exact match in snippet view article find links to article

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 same

Scientific method (23,312 words) [view diff] exact match in snippet view article find links to article

mathematicians, of Euler's formula for polyhedra. H.S.M. Coxeter (1973) Regular Polytopes ISBN 9780486614809, Chapter IX "Poincaré's proof of Euler's formula" "Charles

Platonic solid (5,645 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 many

In Pursuit of the Unknown (809 words) [view diff] exact match in snippet view article find links to article

root of minus one i 2 = − 1 {\displaystyle \mathrm {i} ^{2}=-1} 6 Euler's formula for polyhedra F − E + V = 2 {\displaystyle F-E+V=2} 7 Normal distribution

Imre Lakatos (4,994 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: Firstly

Augustin-Louis Cauchy (5,401 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 important

Eberhard's theorem (1,380 words) [view diff] exact match in snippet view article find links to article

incident to exactly four edges. In this case the equation derived from Euler's formula is not affected by the number p 4 {\displaystyle p_{4}} of quadrilaterals

Incircle and excircles (5,710 words) [view diff] exact match in snippet view article find links to article

58-61. Johnson 1929, p. 187. Emelyanov, Lev, and Emelyanova, Tatiana. "Euler's formula and Poncelet's porism", Forum Geometricorum 1, 2001: pp. 137–140. Josefsson

Four color theorem (6,277 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 degree

Dessin d'enfant (4,171 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 transformation

Spin (physics) (10,584 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 R ^ ( θ , n ^ ) = e i θ 2 n ^ ⋅ σ =

Cycle basis (3,322 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}

Riemann zeta function (10,674 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 Πp ⁠p/p − 1⁠) implies that there are infinitely many

Small cancellation theory (4,247 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 is

Electric dipole transition (2,569 words) [view diff] exact match in snippet view article find links to article

inhomogeneous term in cosines. This can easily be solved by using the Euler's formula for the cosine. We get the following solution: ρ e g ( t ) = Ω / 2

Lemniscate constant (5,924 words) [view diff] exact match in snippet view article find links to article

π can be developed using trigonometric angle sum identities, e.g. Euler's formula 1 4 π = arctan ⁡ 1 2 + arctan ⁡ 1 3 {\textstyle {\tfrac {1}{4}}\pi

Bernoulli number (13,056 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 another

Continued fraction (8,708 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 the

Quadratic reciprocity (8,566 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. {\displaystyle

Planar separator theorem (10,072 words) [view diff] exact match in snippet view article find links to article

with no vertex having degree greater than three. From a corollary of Euler's formula, the number of vertices in the resulting graph will be n ≤ 6 n 0 −

Lemniscate elliptic functions (23,805 words) [view diff] exact match in snippet view article find links to article

π can be developed using trigonometric angle sum identities, e.g. Euler's formula 1 4 π = arctan ⁡ 1 2 + arctan ⁡ 1 3 {\textstyle {\tfrac {1}{4}}\pi