#: Consider a unit sphere placed at the origin, a rotation around the x, y or z axis will map the sphere onto itself, indeed any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis, see Euler angles.

Euler proved that the projections of the angular velocity pseudovector over these three axes was the derivative of its associated angle ( which is equivalent to decompose the instant rotation in three instantaneous Euler rotations ).

In the case of 3 × 3 matrices, three such rotations suffice ; and by fixing the sequence we can thus describe all 3 × 3 rotation matrices ( though not uniquely ) in terms of the three angles used, often called Euler angles.

In the same way the hyperspherical space of 3D rotations can be parameterized by three angles ( Euler angles ), but any such parameterization is degenerate at some points on the hypersphere, leading to the problem of gimbal lock.

The body orientation may be defined by three Euler angles, the Tait-Bryan rotations, a quaternion, or a direction cosine matrix ( rotation matrix ).

However, it is often desirable to represent rotations by a set of three numbers, known as Euler angles ( in numerous variants ), both because this is conceptually simpler, and because one can build a combination of three gimbals to produce rotations in three dimensions.

In formal language, gimbal lock occurs because the map from Euler angles to rotations ( topologically, from the 3-torus T < sup > 3 </ sup > to the real projective space RP < sup > 3 </ sup >) is not a covering map – it is not a local homeomorphism at every point, and thus at some points the rank ( degrees of freedom ) must drop below 3, at which point gimbal lock occurs.

Euler angles provide a means for giving a numerical description of any rotation in three dimensional space using three numbers, but not only is this description not unique, but there are some points where not every change in the target space ( rotations ) can be realized by a change in the source space ( Euler angles ).

This can be explained intuitively by the fact that a quaternion describes a rotation in one single move (" please turn radians around the axis driven by vector "), while the Euler angles are made of three successive rotations.

Euler rotations of the Earth.

Euler rotations are a set of three rotations defined as the movement obtained by changing one of the Euler angles while leaving the other two constant.

Euler rotations are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture.

* Euler angles ( θ, φ, ψ ), representing a product of rotations about the z -, y-and z-axes ;

Euler angles also represent three composed rotations that move a reference frame to a given referred frame.

Euler angles are a means of representing the spatial orientation of any frame ( coordinate system ) as a composition of rotations from a frame of reference ( coordinate system ).

When Euler Angles are defined as a sequence of rotations all the solutions can be valid, but there will be only one inside the angle ranges.

According with these equivalences, proper Euler angles are equivalent to three combined rotations repeating exactly one axis.

When the intrinsic rotations equivalence is used to name the possible conventions of Euler Angles the names are like Z-X ’- Z ’’.

While proper Euler angles are equivalent to three combined rotations repeating exactly one axis, Tait – Bryan angles are equivalent to three composed rotations in different axes.

Following the classical dynamics of Newton and Euler, the motion of a material body is produced by the action of externally applied forces which are assumed to be of two kinds: surface forces and body forces.

If objects are seen as moving within a rotating frame, this movement results in another fictitious force, the Coriolis force ; and if the rate of rotation of the frame is changing, a third fictitious force, the Euler force is experienced.

The reason why Euler and some other authors relate the Cauchy – Riemann equations with analyticity is that a major theorem in complex analysis says that holomorphic functions are analytic and viceversa.

* In the plane ( d = 2 ), if there are b vertices on the convex hull, then any triangulation of the points has at most 2n − 2 − b triangles, plus one exterior face ( see Euler characteristic ).

The standard equations of inviscid flow are the Euler equations.

) A vortex flow of any strength may be added to this uniform flow and the equation is solved, thus there are many flows that solve the Euler equations.

While Maxwell's equations are consistent within special and general relativity, there are some quantum mechanical situations in which Maxwell's equations are significantly inaccurate: including extremely strong fields ( see Euler – Heisenberg Lagrangian ) and extremely short distances ( see vacuum polarization ).

If the generalized coordinates are represented as a vector and time differentiation is represented by a dot over the variable, then the equations of motion ( known as the Lagrange or Euler – Lagrange equations ) are a set of equations:

His work is notable for the use of the zeta function ζ ( s ) ( for real values of the argument " s ", as are works of Leonhard Euler, as early as 1737 ) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π ( x )/( x / ln ( x )) as x goes to infinity exists at all, then it is necessarily equal to one.

The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms.

where, by definition, the left hand side is ζ ( s ) and the infinite product on the right hand side extends over all prime numbers p ( such expressions are called Euler products ):

The Euler product formula can be used to calculate the asymptotic probability that s randomly selected integers are set-wise coprime.

In number theory, the Euler numbers are a sequence E < sub > n </ sub > of integers defined by the following Taylor series expansion:

The odd-indexed Euler numbers are all zero.

The Madelung equations, being quantum Euler equations ( fluid dynamics ), differ philosophically from the de Broglie – Bohm mechanics and are the basis of the hydrodynamic interpretation of quantum mechanics.

In classical mechanics, the equation of motion is Newton's second law, and equivalent formulations are the Euler – Lagrange equations and Hamilton's equations.

Venn diagrams are similar to Euler diagrams.

This means that as the number of contours increases, Euler diagrams are typically less visually complex than the equivalent Venn diagram, particularly if the number of non-empty intersections is small.

Venn diagrams are very similar to Euler diagrams, which were invented by Leonhard Euler ( 1708 – 1783 ) in the 18th century.

In number theory, Euler's theorem ( also known as the Fermat – Euler theorem or Euler's totient theorem ) states that if n and a are coprime positive integers, then

In the shortest of them ( 43 pages as of 2009 ), which he titles " Apology for the Proof of the Riemann Hypothesis " ( using the word " apology " in the rarely used sense of apologia ), he claims to use his tools on the theory of Hilbert spaces of entire functions to prove the Riemann Hypothesis for Dirichlet L-functions ( thus proving GRH ) and a similar statement for the Euler zeta function, and even to be able to assert that zeros are simple.

It can be defined as a change in direction of the rotation axis in which the second Euler angle ( nutation ) is constant.

Alternatively, it can be defined in terms of the Euler characteristic χ, via the relationship χ = 2 − 2g for closed surfaces, where g is the genus.

Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k, where k is the non-orientable genus.

The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids.

The Euler characteristic was classically defined for the surfaces of polyhedra, according to the formula

Along a different line of study, there is a combinatorially defined cohomology theory of knots called Khovanov homology whose Euler characteristic is the Jones polynomial.

Khovanov and Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical invariants.

Gauss defined primitive roots in Article 57 of the Disquisitiones Arithmeticae ( 1801 ), where he credited Euler with coining the term.

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by

The Gauss map can be defined ( globally ) if and only if the surface is orientable, in which case its degree is half the Euler characteristic.

Euler angles between two frames are defined only if both frames have the same handedness.

An L-function L ( E, s ) can be defined for an elliptic curve E by constructing an Euler product from the number of points on the curve modulo each prime p. This L-function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form.

Euler generalised the problem considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper On the Number of Primes Less Than a Given Magnitude, in which he defined his zeta function and proved its basic properties.

Every rotation in three dimensions is defined by its axis — a direction that is left fixed by the rotation — and its angle — the amount of rotation about that axis ( Euler rotation theorem ).

The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron.

: Here the " Euler factor " P ( τ | B ; x ) is defined to be the element det ( 1-τx | B ) considered as an element of O, which when x happens to act on B is not the same as det ( 1-τx | B ) considered as an element of O.

The Euler characteristic of a sheaf is defined by