The coordinate systems chosen for atomic orbitals are usually spherical coordinates ( r, θ, φ ) in atoms and cartesians ( x, y, z ) in poly-atomic molecules.

The advantage of spherical coordinates ( for atoms ) is that an orbital wave function is a product of three factors each dependent on a single coordinate: ψ ( r, θ, φ ) = R ( r ) Θ ( θ ) Φ ( φ ).

On the plane the most common alternative is polar coordinates, where every point is represented by its radius r from the origin and its angle θ.

where θ < sub > r, p </ sub > is the angle between r and p measured from r to p ; an important distinction because without it, the sign of the cross product would be meaningless.

The direction of u < sub > r </ sub > is described by θ, the angle between the x-axis and the unit vector, measured counterclockwise from the x-axis.

The other unit vector for polar coordinates, u < sub > θ </ sub > is perpendicular to u < sub > r </ sub > and points in the direction of increasing θ.

These polar unit vectors can be expressed in terms of Cartesian unit vectors in the x and y directions, denoted i and j respectively :< ref > Note: unlike the Cartesian unit vectors i and j, which are constant, in polar coordinates the direction of the unit vectors u < sub > r </ sub > and u < sub > θ </ sub > depend on θ, and so in general have non-zero time derivatives .</ ref >

Polar unit vectors at two times t and t + dt for a particle with trajectory r ( t ); on the left the unit vectors u < sub > ρ </ sub > and u < sub > θ </ sub > at the two times are moved so their tails all meet, and are shown to trace an arc of a unit radius circle.

By moving the unit vectors so their tails coincide, as seen in the circle at the left of the image above, it is seen that u < sub > ρ </ sub > and u < sub > θ </ sub > form a right-angled pair with tips on the unit circle that trace back and forth on the perimeter of this circle with the same angle θ ( t ) as r ( t ).

To remain orthogonal to u < sub > ρ </ sub > while the trajectory r ( t ) rotates an amount dθ, u < sub > θ </ sub >, which is orthogonal to r ( t ), also rotates by dθ.

For trajectories other than circular motion, for example, the more general trajectory envisioned in the image above, the instantaneous center of rotation and radius of curvature of the trajectory are related only indirectly to the coordinate system defined by u < sub > ρ </ sub > and u < sub > θ </ sub > and to the length | r ( t )|

The following equation on the polar coordinates ( r, θ ) describes a general ellipse with semidiameters a and b, centered at a point ( r < sub > 0 </ sub >, θ < sub > 0 </ sub >), with the a axis rotated by φ relative to the polar axis:

In general, this can be recognized as the equation of a conic section in polar coordinates ( r, θ ).

If an object moves with angular velocity ω around a circle of radius r centered at the origin of the x-y plane, then its motion along each coordinate is simple harmonic motion with amplitude r and angular frequency ω.

In the plane perpendicular to that axis, the tidal acceleration is directed inwards ( towards the center where ∆ r is zero ), and its magnitude is ( axial ) in linear approximation as in Figure 2.

In the second form given above, the phase is often generalized to, by replacing the wavenumber k with a wave vector that specifies the direction and wavenumber of a plane wave in 3-space, parameterized by position vector r. In that case, the wavenumber k, the magnitude of k, is still in the same relationship with wavelength as shown above, with v being interpreted as scalar speed in the direction of the wave vector.

The power P passing through a circle of radius r in the transverse plane at position z is

Schrödinger required that a wave packet solution ( not just plane waves ) at position r with wavevector k will move along the trajectory determined by classical mechanics in the limit that the wavelength is small, i. e. for a large k and therefore large p in comparison to Planck's reduced constant ħ. Equivalently, in the limit as ħ approaches zero, the equations of classical mechanics are restored from quantum mechanics.

surface's tangent plane ; and the geodesic torsion ( or relative torsion ), τ < sub > r </ sub >, measures the rate of change of the surface normal around the curve's tangent.

These are the kinematic equations for a particle traversing a path in a plane, described by position r = r ( t ).

The idea is that a point with position vector is in the plane if and only if the vector drawn from to is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be expressed as the set of all points r such that

The spectral radius, r ( T ), of T is the radius of the smallest circle in the complex plane which is centered at the origin and contains the spectrum σ ( T ) inside of it, i. e.

In geometrical terms, given a circle centered about the origin in the plane with radius r, the problem asks how many integer lattice points lie on or inside the circle.

constant ) and let the position of a particle moving in this plane be described with the remaining Schwarzschild coordinates ( r, φ ).

Then replace the ( r, φ ) plane with a surface dimpled in the w direction according to the equation ( Flamm's paraboloid )

Therefore, the displacement vector r and its velocity v are always in the plane perpendicular to the constant vector L.

In the plane, the inverse of a point P with respect to a reference circle of center O and radius r is a point P, lying on the ray from O through P such that

Therefore, letting r tend to infinity ( we let r tend to infinity since f is analytic on the entire plane ) gives a < sub > k </ sub >

where ε < sub > r </ sub > is the dielectric constant of the dispersion medium, ε < sub > 0 </ sub > is the permittivity of free space ( C² N < sup >− 1 </ sup > m < sup >− 2 </ sup >), η is dynamic viscosity of the dispersion medium ( Pa s ), and ζ is zeta potential ( i. e., the electrokinetic potential of the slipping plane in the double layer ).

For non-constant meromorphic functions in the plane, T ( r, f ) tends to infinity as r tends to infinity,

By the First Fundamental Theorem, 0 ≤ δ ( a, f ) ≤ 1, if T ( r, f ) tends to infinity ( which is always the case for non-constant functions meromorphic in the plane ).

A circle of radius r for the Chebyshev distance ( L < sub >∞</ sub > metric ) on a plane is also a square with side length 2r parallel to the coordinate axes, so planar Chebyshev distance can be viewed as equivalent by rotation and scaling to planar taxicab distance.

And to " measure " is to place a shorter measuring length s successively ( q times ) along longer length l until the remaining portion r is less than the shorter length s. In modern words, remainder r = l − q * s, q being the quotient, or remainder r is the " modulus ", the integer-fractional part left over after the division.

In most varieties of North American English, the sound corresponding to the letter r is an alveolar approximant or retroflex rather than a trill or a tap.

The loss of syllable-final r in North America is confined mostly to the accents of eastern New England, New York City and surrounding areas and the coastal portions of the South, and African American Vernacular English.

In rural tidewater Virginia and eastern New England, ' r ' is non-rhotic in accented ( such as " bird ", " work ", " first ", " birthday ") as well as unaccented syllables, although this is declining among the younger generation of speakers.

Dropping of syllable-final r sometimes happens in natively rhotic dialects if r is located in unaccented syllables or words and the next syllable or word begins in a consonant.

* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

The equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = r < sup > 2 </ sup > is the equation for any circle with a radius of r.

* Most values beyond 3. 57 exhibit chaotic behaviour, but there are still certain isolated ranges of r that show non-chaotic behavior ; these are sometimes called islands of stability.

* The development of the chaotic behavior of the logistic sequence as the parameter r varies from approximately 3. 5699 to approximately 3. 8284 is sometimes called the Pomeau – Manneville scenario, which is characterized by a periodic ( laminar ) phase interrupted by bursts of aperiodic behavior.

Some of the more important of these include the r process, which involves rapid neutron captures, the rp process, which involves rapid proton captures, and the p process ( sometimes known as the gamma process ), which involves photodisintegration of existing nuclei.

The linear permittivity of a homogeneous material is usually given relative to that of free space, as a relative permittivity ε < sub > r </ sub > ( also called dielectric constant, although this sometimes only refers to the static, zero-frequency relative permittivity ).

* a phoneme usually represented by / r ̂/ ( sometimes written dr ) that was possibly an alveolar tap, though some have argued for an aspirated affricate.

The guttural r sound can, however, still sometimes be detected, especially amongst elderly populations in more rural areas.

Compensatory lengthening of vowel before cluster of sonorant ( r, l, n, m, w, sometimes y ) and s, after deletion of s. ~ Aeolic: compensatory lengthening of sonorant.

In later uncial scripts, the letters are sometimes drawn haphazardly ; for example, double-l runs together at the baseline, bows ( for example in b, p, r ) do not entirely curve in to touch their stems, and the script is generally not written as cleanly as previously.

Applied behavior analysis, which is the name of the discipline directly descended from Skinner's work, holds that behavior is explained in four terms: conditioned stimulus ( S < sup > C </ sup >), a discriminative stimulus ( S < sup > d </ sup >), a response ( R ), and a reinforcing stimulus ( S < sup > rein </ sup > or S < sup > r </ sup > for reinforcers, sometimes S < sup > ave </ sup > for aversive stimuli ).

: ε < sub > r </ sub > is the relative static permittivity ( sometimes called the dielectric constant ) of the material between the plates ( for a vacuum, );

Since there is a mark of the Qing Dynasty ruler Qianlong ( r. 1735 – 1796 ) of China, dated 1770, this Korean-designed rain gauge is sometimes misunderstood as having been imported from China.

The term is sometimes used to refer to the Fraser Canyon and stretches upstream from there, but in general British Columbian usage of the term refers to the stretch of the river downstream from the town of Hope ( about where the " r " in " Vancouver " is on the map below ).

In statistics, the Pearson product-moment correlation coefficient ( sometimes referred to as the PPMCC or PCC, or Pearson's r, and is typically denoted by r ) is a measure of the correlation ( linear dependence ) between two variables X and Y, giving a value between + 1 and − 1 inclusive.

They were sometimes called the Tenerife Goldcrest, no matter which of the islands they lived on ; however, a 2006 study of the vocalisations of these birds indicate that they actually comprise two subspecies of the Goldcrest that are separable on voice ; R. r. teneriffae occurring on Tenerife and the newly described subspecies, R. r. ellenthalerae, occurring on the smaller islands of La Palma and El Hierro.

The next term that becomes significant is proportional to 1 / r < sup > 2 </ sup > and is sometimes called the induction term.

For even smaller r, terms proportional to 1 / r < sup > 3 </ sup > become significant ; this is sometimes called the electrostatic field term and can be thought of as stemming from the electrical charge in the antenna element.

Rather, in the near-field, it is sometimes useful to express the contributions as a sum of radiating fields combined with evanescent fields, where the latter are exponentially decaying with r. And in the source itself, or as soon as one enters a region of inhomogeneous materials, the multipole expansion is no longer valid and the full solution of Maxwell's equations is generally required.

It is also sometimes convenient to split the conversion factor β < sub > r </ sub > into a " macroscopic part " that relates to the overall geometry of the emitter and its surroundings, and a " local part " that relates to the ability of the very-local structure of the emitter surface to enhance the electric field.

They were sometimes called the Tenerife Goldcrest, no matter which of the islands they lived on ; however, a 2006 study of the vocalisations of these birds indicate that they actually comprise two subspecies of the Goldcrest that are separable on voice ; R. r. teneriffae occurring on Tenerife and the newly described subspecies, R. r. ellenthalerae, occurring on the smaller islands of La Palma and El Hierro.