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

searching for Euler's formula 58 found (315 total)

alternate case: euler's formula

Euler characteristic (3,271 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
Planar 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 is
Johnson'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 the
Johann 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 Berlin
Polyhedral 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 coefficients
Euler'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 that
Lewis' 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 display
Polyhedron (7,432 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 or
Euler'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 with
Sine 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 ) sin
Cuboid (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 convex
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 edges
Lazy caterer's sequence (715 words) [view diff] exact match in snippet view article find links to article
mathworld.wolfram.com. Retrieved 2020-08-11. Moore, T. L. (1991), "Using Euler's formula to solve plane separation problems", The College Mathematics Journal
Viète'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 ⁡
Spiral 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 function
Kepler–Poinsot polyhedron (2,190 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
Eulerian 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 lattice
Simplicial 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 2n
Euler 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 faces
Stacked 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 true
Kuratowski'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 into
Fáry'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 defines
Münchenstein 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 loads
Grinberg'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 only
Jacket 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 + i
Discharging 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 6
Polytope (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 alternating
Mac 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 circuit
Crossing number inequality (1,340 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
Thomas 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 each
Murderous Maths (2,024 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
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}
Occurrences 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 +
Automedian triangle (1,459 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
Jurij 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 ) , {\displaystyle
Hidden-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) faces
Francesco Maurolico (1,579 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
Graph theory (6,182 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 (2,118 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 same
Platonic solid (5,033 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
Heegner 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 equivalent
Euler's critical load (1,863 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 compressive
Imre Lakatos (4,888 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
Euler's totient function (6,068 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 multiplicative
Incircle and excircles of a triangle (5,520 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-Court
Four color theorem (5,597 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 (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 transformation
Augustin-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 important
Spin (physics) (10,210 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 ) +
Scientific method (17,028 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 refutations
Riemann zeta function (9,255 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 primes
Cycle 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 is
Bernoulli number (11,886 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
Generalized continued fraction (8,683 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 (9,201 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 (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