History.

The city was founded in 1265 by King Ottokar II of Bohemia, who granted its municipal charter in 1265. The siting and planning of the city was carried out by the king's knight Hirzo. The settlers were coming from the Bohemian Forest and Upper Austria. The royal city was created as a platform of the king's power in South Bohemia and to counterbalance the powerful noble House of Rosenberg, which became extinct in 1611.  In 1341 King John of Bohemia allowed Jewish families to reside within the city walls, and the first synagogue was built in 1380; however several pogroms occurred in the late 15th and early 16th century. Since the Hussite Wars, the city was traditionally a bulwark of the Catholic Church during the long-lasting religious conflicts in the Kingdom of Bohemia. A part of the Habsburg Monarchy from 1526, Budejovice remained a loyal supporter of Emperor Ferdinand II in the Thirty Years' War. Budějovice underwent a short occupation by Prussia during the Silesian Wars, and the war between the Habsburgs and the French army in 1742.  In 1762 the Piarists established a "gymnasium" here and Emperor Joseph II founded the diocese in 1785. In 1847, the production of Koh-i-Noor Hardtmuth pencils was relocated from Vienna to Budějovice.  During the Second World War in March 1945, Budějovice was twice targeted by U.S. Air Force raids that greatly damaged the city and caused great loss of life. At the end of the war, on 9 May 1945, Soviet troops liberated the city. On the following day, the Red Army and the American Army met on the main square in a joint celebration of the city's liberation.

Demographics.

The vast majority of the city's population today are Czechs (94.9% in 2001), with 1.15% Slovaks. In the past, the city had a significant proportion of ethnic Germans, who had formed the majority since medieval times. The city remained a German-speaking enclave until 1880, after which Czechs became the majority. Until the end of World War II, the city contained a significant German minority (about 15.5% in 1930). For example, the ratios between the Germans and the Czechs were in 1880: 11,829 Germans to 11,812 Czechs, in 1890: 11,642 to 16,585, in 1900: 15,400 to 23,400, in 1910: 16,900 to 27,300 and in 1921: 7,415 to 35,800. The entire German population was forcibly expelled in 1945 under the Beneš decrees.

Geography.

České Budějovice is a low-lying city spread mostly across a plain, making it nearly flat in the inner parts with hillier areas in the eastern suburbs. The lowest point lies at above sea level, and the highest point at. Because it is not very well ventilated some strong winters do occur; the strongest winter plummeted to in 1929 in the southern part of the city with lower temperatures elsewhere in the meantime. Nevertheless, such a strong winter is exceptional, especially outside of valley bottoms.  České Budějovice has a cooler and wet inland version of a humid continental climate ("Dfb") with an average annual temperature of. There are four seasons, with a murky dry winter between early December and early March, a sunny and wetter spring between half of March up to half of May changing to a rainy and warm summer during late May and early September when a dry autumn lasting to late November begins. There are between 1550 and 1800 hours of sunshine in most years.

Beer.

Budějovice has long been well known for the beer brewed there since the 13th century. For a time, the town was the imperial brewery for the Holy Roman Emperor, and "Budweiser Bier" (i.e. beer from Budweis) became, along with "Pilsner" from Plzeň, one of the best-known lagers. Brewing remains a major industry. In 1256 the Svitavy brewery was founded there, which was closed in 2002.  The largest brewery, founded in 1895, is "Pivovar Budějovický Budvar" (Budweiser Budvar Brewery) which has legal rights to market its beer under the "Budweiser" brand name in much of Europe. The same product is also sold elsewhere under the names "Budvar" and "Czechvar" due to legal disagreements with Anheuser-Busch over the Budweiser brand and Anheuser-Busch sells its beer as "Bud" in most of the European Union.. The American lager was originally brewed as an imitation of the famous Bohemian original, but over time has developed its own identity and attained remarkable commercial success. Anheuser-Busch has made offers to buy out the Czech brewing company in order to secure global rights to the name "Budweiser", but the Czech government has refused all such offers, regarding the Czech Budweiser name as a matter of national pride.  The oldest (founded in 1795) and second largest brewery was renamed to "Pivovar Samson", replacing its original German name "Budweiser Bürgerbräu" during the communist period. It also exported, mostly under the "Samson" and "Crystal" labels. Recently, they reacquired naming rights for Budweiser for Europe while offering "B. B. Bürgerbräu" in the US since 2005.

Architecture.

The old town preserves interesting architecture from the Gothic, Renaissance, Baroque, and 19th century periods. This includes buildings around the large Ottokar II Square ("Náměstí Přemysla Otakara II"), the old Town Hall with murals and bronze gargoyles, and the 16th century Black Tower ("Černá věž"). The most valuable historic building in České Budějovice is the Dominican convent with the Gothic Presentation of the Virgin Mary church from the 13th century. The Iron Maiden Tower and the Rabenštejn Tower are a 14th-century former prisons and one of the few remainings of the Old Town's Gothic fortifications.

Culture.

The Museum of South Bohemia dates to 1877 and holds a large collection of historic books, coins, weapons and other articles. It was closed for reconstruction in 2012–2014.

In literature.

The city is one of the major settings in the novel "The Good Soldier Švejk" by Jaroslav Hašek. Budějovice is the setting and was the working title for the play The Misunderstanding by Albert Camus.

Transport.

The city can be reached from other locations by inter-city buses and by train. The town will receive access to the planned D3 motorway running from Prague to the Austrian border at Dolní Dvořiště. Internationally, a direct railroad built by the Austrian Empress Elisabeth Railway company in 1871, connecting the Czech capital Prague with Zürich, via Linz and Salzburg, also makes a stop in České Budějovice.  The city is served by České Budějovice railway station, a Neo-Renaissance style station building in the new town. The horse-drawn railroad line connecting České Budějovice to Linz was the second oldest public line in continental Europe (after the St.Étienne-Andrézieux line in France), constructed from 1824 to 1832; traces of the line can be seen south of the city. Local buses and trolleybuses take passengers to most areas of the city.  Public domestic and non-public international České Budějovice Airport is located south-west from České Budějovice, at the nearby village of Planá. Modernization of the airport will allow public international flights, construction of the new terminal started in December 2017 and full operation is planned at the end of 2020.

Sport.

České Budějovice is the site of many sports facilities and national stadiums. For example, the football Stadion Střelecký ostrov, the hockey Budvar Arena and the Athletic Stadium Sokol. The Swimming Stadium České Budějovice features a 50-meter indoor pool, a diving pool, saunas, an outdoor swimming pool and a children's pool. After the modernization in 1998 a covered water slide was added and after the modernization in 2017 a new whirlpool. Major sport clubs include:

Twin towns – sister cities.

České Budějovice is twinned with:

České Budějovice (; ; ) is a statutory city in the Czech Republic. It is the largest city in the South Bohemian Region as well as its political and commercial capital, the seat of the Roman Catholic Diocese of České Budějovice, the University of South Bohemia, and the Academy of Sciences. It is located in the center of a valley of the Vltava River, at the confluence with the Malše. It is famous for Budweiser.

České Budějovice (,, popřípadě ) jsou statutární město a správní a kulturní metropole Jihočeského kraje. Leží v Českobudějovické pánvi na soutoku řek Vltava a Malše, mají obyvatel a nachází se v nich řada historických památek a muzeí. Pro ně a blízkost dalších historicky cenných míst (Hluboká nad Vltavou, Český Krumlov, Zlatá Koruna, Vyšší Brod, Třeboň atd.) jsou častým cílem turistů. Ve městě sídlí Jihočeská univerzita v Českých Budějovicích a biskup římskokatolické českobudějovické diecéze. Historické centrum města bylo vyhlášeno 19. března 1980 městskou památkovou rezervací.


Overview.

The 'enthalpy' of fusion is a latent heat, because during melting the heat energy needed to change the substance from solid to liquid at atmospheric pressure is latent heat of fusion, as the temperature remains constant during the process. The latent heat of fusion is the enthalpy change of any amount of substance when it melts. When the heat of fusion is referenced to a unit of mass, it is usually called the specific heat of fusion, while the molar heat of fusion refers to the enthalpy change per amount of substance in moles.  The liquid phase has a higher internal energy than the solid phase. This means energy must be supplied to a solid in order to melt it and energy is released from a liquid when it freezes, because the molecules in the liquid experience weaker intermolecular forces and so have a higher potential energy (a kind of bond-dissociation energy for intermolecular forces).  When liquid water is cooled, its temperature falls steadily until it drops just below the line of freezing point at 0 °C. The temperature then remains constant at the freezing point while the water crystallizes. Once the water is completely frozen, its temperature continues to fall.  The enthalpy of fusion is almost always a positive quantity; helium is the only known exception. Helium-3 has a negative enthalpy of fusion at temperatures below 0.3 K. Helium-4 also has a very slightly negative enthalpy of fusion below. This means that, at appropriate constant pressures, these substances freeze with the addition of heat. In the case of He, this pressure range is between 24.992 and.  These values are mostly from the CRC "Handbook of Chemistry and Physics", 62nd edition. The conversion between cal/g and J/g in the above table uses the thermochemical calorie (cal) = 4.184 joules rather than the International Steam Table calorie (cal) = 4.1868 joules.

Examples.

A) To heat 1 kg (1.00 liter) of water from 283.15 K to 303.15 K (10 °C to 30 °C) requires 83.6 kJ. However, to melt ice also requires energy. We can treat these two processes independently; thus, to heat 1 kg of ice from 273.15 K to water at 293.15 K (0 °C to 20 °C) requires:  From these figures it can be seen that one part ice at 0 °C will cool almost exactly 4 parts water from 20 °C to 0 °C.  B) Silicon has a heat of fusion of 50.21 kJ/mol. 50 kW of power can supply the energy required to melt about 100 kg of silicon in one hour, after it is brought to the melting point temperature:  50 kW = =  /h * (1 mol Si)/ * /(mol Si) * / =

Solubility prediction.

The heat of fusion can also be used to predict solubility for solids in liquids. Provided an ideal solution is obtained the mole fraction formula_1 of solute at saturation is a function of the heat of fusion, the melting point of the solid formula_2 and the temperature (T) of the solution:  Here, R is the gas constant. For example, the solubility of paracetamol in water at 298 K is predicted to be:  This equals to a solubility in grams per liter of:  formula_5  which is a deviation from the real solubility (240 g/L) of 11%. This error can be reduced when an additional heat capacity parameter is taken into account.

Proof.

At equilibrium the chemical potentials for the pure solvent and pure solid are identical:  or  with formula_8 the gas constant and formula_9 the temperature.  Rearranging gives:  and since  the heat of fusion being the difference in chemical potential between the pure liquid and the pure solid, it follows that  Application of the Gibbs–Helmholtz equation:  ultimately gives:  or:  and with integration:  the end result is obtained:

The enthalpy of fusion of a substance, also known as (latent) heat of fusion is the change in its enthalpy resulting from providing energy, typically heat, to a specific quantity of the substance to change its state from a solid to a liquid, at constant pressure. For example, when melting 1 kg of ice (at 0 °C under a ), 333.55 kJ of energy is absorbed with no temperature change. The heat of solidification (when a substance changes from liquid to solid) is equal and opposite.

Termoska (též termoláhev) je speciálně vyrobená láhev, která slouží k uchování teploty obsahu. Například teplého nebo studeného nápoje (pokrmu). V roce 1892 ji vynalezl pracovník Oxfordské univerzity Sir James Dewar, který potřeboval pro účely laboratorních pokusů nádobu, která by po dlouhou dobu udržela velmi chladné tekutiny. V laboratorním a průmyslovém použití se používá název Dewarova nádoba nebo i "dewarka". Používá se například k uchování kapalného dusíku.


Example.

In the example illustrated to the right (or above in some mobile versions), a particle or body P is at a fixed distance "r" from the origin, "O", rotating counterclockwise. It becomes important to then represent the position of particle P in terms of its polar coordinates ("r", "θ"). In this particular example, the value of "θ" is changing, while the value of the radius remains the same. (In rectangular coordinates ("x", "y") both "x" and "y" vary with time). As the particle moves along the circle, it travels an arc length "s", which becomes related to the angular position through the relationship:-

Measurements.

Angular displacement may be measured in radians or degrees. Using radians provides a very simple relationship between distance traveled around the circle and the distance "r" from the centre.  For example, if a body rotates 360° around a circle of radius "r", the angular displacement is given by the distance traveled around the circumference - which is 2π"r" - divided by the radius: formula_3 which easily simplifies to: formula_4. Therefore, 1 revolution is formula_5 radians. When a particle travels from point P to point Q over formula_6, as it does in the illustration to the left, the radius of the circle goes through a change in angle formula_7 which equals the "angular displacement".

Three dimensions.

In three dimensions, angular displacement is an entity with a direction and a magnitude. The direction specifies the axis of rotation, which always exists by virtue of the Euler's rotation theorem; the magnitude specifies the rotation in radians about that axis (using the right-hand rule to determine direction). This entity is called an axis-angle.  Despite having direction and magnitude, angular displacement is not a vector because it does not obey the commutative law for addition. Nevertheless, when dealing with infinitesimal rotations, second order infinitesimals can be discarded and in this case commutativity appears.  Several ways to describe angular displacement exist, like rotation matrices or Euler angles. See charts on SO(3) for others.

Matrix notation.

Given that any frame in the space can be described by a rotation matrix, the displacement among them can also be described by a rotation matrix. Being formula_8 and formula_9 two matrices, the angular displacement matrix between them can be obtained as formula_10. When this product is performed having a very small difference between both frames we will obtain a matrix close to the identity.  In the limit, we will have an infinitesimal rotation matrix.

Infinitesimal rotation matrices.

An infinitesimal angular displacement is an infinitesimal rotation matrix:  We can introduce here the infinitesimal angular displacement tensor or rotation generator associated:  Such that its associated rotation matrix is formula_13. When it is divided by the time, this will yield the angular velocity vector.

Generators of rotations.

Suppose we specify an axis of rotation by a unit vector ["x", "y", "z"], and suppose we have an infinitely small rotation of angle Δθ about that vector. Expanding the rotation matrix as an infinite addition, and taking the first order approach, the rotation matrix Δ"R" is represented as:  A finite rotation through angle θ about this axis may be seen as a succession of small rotations about the same axis. Approximating Δθ as θ/"N" where "N" is a large number, a rotation of θ about the axis may be represented as:  It can be seen that Euler's theorem essentially states that all rotations may be represented in this form. The product formula_16 is the "generator" of the particular rotation, being the vector ("x","y","z") associated with the matrix A. This shows that the rotation matrix and the axis-angle format are related by the exponential function.  One can derive a simple expression for the generator G. One starts with an arbitrary plane defined by a pair of perpendicular unit vectors a and b. In this plane one can choose an arbitrary vector x with perpendicular y. One then solves for y in terms of x and substituting into an expression for a rotation in a plane yields the rotation matrix R which includes the generator G = ba − ab.  To include vectors outside the plane in the rotation one needs to modify the above expression for R by including two projection operators that partition the space. This modified rotation matrix can be rewritten as an exponential function.  Analysis is often easier in terms of these generators, rather than the full rotation matrix. Analysis in terms of the generators is known as the Lie algebra of the rotation group.

Relationship with Lie algebras.

The matrices in the Lie algebra are not themselves rotations; the skew-symmetric matrices are derivatives, proportional differences of rotations. An actual "differential rotation", or "infinitesimal rotation matrix" has the form where is vanishingly small and, for instance with,  The computation rules are as usual except that infinitesimals of second order are routinely dropped. With these rules, these matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals. It turns out that "the order in which infinitesimal rotations are applied is irrelevant". To see this exemplified, consult infinitesimal rotations SO(3).

Exponential map.

Connecting the Lie algebra to the Lie group is the exponential map, which is defined using the standard matrix exponential series for For any skew-symmetric matrix, exp() is always a rotation matrix.  An important practical example is the case. In rotation group SO(3), it is shown that one can identify every with an Euler vector, where is a unit magnitude vector.  By the properties of the identification, is in the null space of. Thus, is left invariant by and is hence a rotation axis.  Using Rodrigues' rotation formula on matrix form with, together with standard double angle formulae one obtains,  where,.  This is the matrix for a rotation around axis by the angle in half-angle form. For full detail, see exponential map SO(3).  Notice that for infinitesimal angles second order terms can be ignored and remains exp(A) = I + A

Angular displacement of a body is the angle in radians (degrees, revolutions) through which a point revolves around a centre or line has been rotated in a specified sense about a specified axis. When a body rotates about its axis, the motion cannot simply be analyzed as a particle, as in circular motion it undergoes a changing velocity and acceleration at any time ("t"). When dealing with the rotation of a body, it becomes simpler to consider the body itself rigid. A body is generally considered rigid when the separations between all the particles remains constant throughout the body's motion, so for example parts of its mass are not flying off. In a realistic sense, all things can be deformable, however this impact is minimal and negligible. Thus the rotation of a rigid body over a fixed axis is referred to as rotational motion.

Úhlová dráha je orientovaný úhel, který opíše průvodič hmotného bodu během jeho pohybu po kružnici. Pojem úhlové dráhy lze ztotožnit s pojmem orientovaného úhlu.


History.

The concepts of angle and radius were already used by ancient peoples of the first millennium BC. The Greek astronomer and astrologer Hipparchus (190–120 BC) created a table of chord functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions. In "On Spirals", Archimedes describes the Archimedean spiral, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system.  From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to Mecca (qibla)—and its distance—from any location on the Earth. From the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. The calculation is essentially the conversion of the equatorial polar coordinates of Mecca (i.e. its longitude and latitude) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the great circle through the given location and the Earth's poles, and whose polar axis is the line through the location and its antipodal point.  There are various accounts of the introduction of polar coordinates as part of a formal coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidge's "Origin of Polar Coordinates." Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs.  In "Method of Fluxions" (written 1671, published 1736), Sir Isaac Newton examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems. In the journal "Acta Eruditorum" (1691), Jacob Bernoulli used a system with a point on a line, called the "pole" and "polar axis" respectively. Coordinates were specified by the distance from the pole and the angle from the "polar axis". Bernoulli's work extended to finding the radius of curvature of curves expressed in these coordinates.  The actual term "polar coordinates" has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in George Peacock's 1816 translation of Lacroix's "Differential and Integral Calculus". Alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them.

Conventions.

The radial coordinate is often denoted by "r" or "ρ", and the angular coordinate by "φ", "θ", or "t". The angular coordinate is specified as "φ" by ISO standard 31-11. However, in mathematical literature the angle is often denoted by θ instead of "φ".  Angles in polar notation are generally expressed in either degrees or radians (2 rad being equal to 360°). Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics.  The angle "φ" is defined to start at 0° from a "reference direction", and to increase for rotations in either counterclockwise (ccw) or clockwise (cw) orientation. In mathematics, e.g., the reference direction is usually drawn as a ray from the pole horizontally to the right, and the polar angle increases to positive angles for ccw rotations, whereas in navigation (bearing, heading) the 0°-heading is drawn vertically upwards and the angle increases for cw rotations. The polar angles decrease towards negative values for rotations in the respectively opposite orientations.

Uniqueness of polar coordinates.

Adding any number of full turns (360°) to the angular coordinate does not change the corresponding direction. Similarly, any polar coordinate is identical to the coordinate with the negative radial component and the opposite direction (adding 180° to the polar angle). Therefore, the same point ("r", "φ") can be expressed with an infinite number of different polar coordinates and, where "n" is an arbitrary integer. Moreover, the pole itself can be expressed as (0, "φ") for any angle "φ".  Where a unique representation is needed for any point besides the pole, it is usual to limit "r" to positive numbers () and "φ" to the interval [0, 360°) or (−180°, 180°] (in radians, [0, 2) or (−, ]). Another convention, in reverence to the usual codomain of the arctan-function, is to allow for arbitrary nonzero real values of the radial component and restrict the polar angle to. In all cases a unique azimuth for the pole ("r" = 0) must be chosen, e.g., "φ" = 0.

Converting between polar and Cartesian coordinates.

The polar coordinates "r" and "φ" can be converted to the Cartesian coordinates "x" and "y" by using the trigonometric functions sine and cosine:  The Cartesian coordinates "x" and "y" can be converted to polar coordinates "r" and "φ" with "r" ≥ 0 and "φ" in the interval (−, ] by:  where atan2 is a common variation on the arctangent function defined as  If "r" is calculated first as above, then this formula for φ may be stated a little more simply using the standard arccosine function:  The value of "φ" above is the principal value of the complex number function arg applied to "x" + "iy". An angle in the range [0, 2) may be obtained by adding 2 to the value in case it is negative (in other words when "y" is negative).

Polar equation of a curve.

The equation defining an algebraic curve expressed in polar coordinates is known as a "polar equation". In many cases, such an equation can simply be specified by defining "r" as a function of "φ". The resulting curve then consists of points of the form ("r"("φ"), "φ") and can be regarded as the graph of the polar function "r". Note that, in contrast to Cartesian coordinates, the independent variable "φ" is the "second" entry in the ordered pair.  Different forms of symmetry can be deduced from the equation of a polar function "r". If the curve will be symmetrical about the horizontal (0°/180°) ray, if it will be symmetric about the vertical (90°/270°) ray, and if it will be rotationally symmetric by α clockwise and counterclockwise about the pole.  Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the polar rose, Archimedean spiral, lemniscate, limaçon, and cardioid.  For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.

Circle.

The general equation for a circle with a center at and radius "a" is  This can be simplified in various ways, to conform to more specific cases, such as the equation for a circle with a center at the pole and radius "a".  When =, or when the origin lies on the circle, the equation becomes  In the general case, the equation can be solved for, giving the solution with a minus sign in front of the square root gives the same curve.

Line.

"Radial" lines (those running through the pole) are represented by the equation where γ is the angle of elevation of the line; that is,, where "m" is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line perpendicularly at the point ("r", γ) has the equation  Otherwise stated ("r", γ) is the point in which the tangent intersects the imaginary circle of radius "r".

Polar rose.

A polar rose is a mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,  for any constant γ (including 0). If "k" is an integer, these equations will produce a "k"-petaled rose if "k" is odd, or a 2"k"-petaled rose if "k" is even. If "k" is rational but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The variable "a" directly represents the length or amplitude of the petals of the rose, while "k" relates to their spatial frequency. The constant γ can be regarded as a phase angle.

Archimedean spiral.

The Archimedean spiral is a spiral that was discovered by Archimedes, which can also be expressed as a simple polar equation. It is represented by the equation Changing the parameter "a" will turn the spiral, while "b" controls the distance between the arms, which for a given spiral is always constant. The Archimedean spiral has two arms, one for and one for. The two arms are smoothly connected at the pole. Taking the mirror image of one arm across the 90°/270° line will yield the other arm. This curve is notable as one of the first curves, after the conic sections, to be described in a mathematical treatise, and as being a prime example of a curve that is best defined by a polar equation.

Conic sections.

A conic section with one focus on the pole and the other somewhere on the 0° ray (so that the conic's major axis lies along the polar axis) is given by:  where "e" is the eccentricity and formula_15 is the semi-latus rectum (the perpendicular distance at a focus from the major axis to the curve). If, this equation defines a hyperbola; if, it defines a parabola; and if, it defines an ellipse. The special case of the latter results in a circle of the radius formula_15.

Intersection of two polar curves.

The graphs of two polar functions formula_17 and formula_18 have possible intersections of three types:

Complex numbers.

Every complex number can be represented as a point in the complex plane, and can therefore be expressed by specifying either the point's Cartesian coordinates (called rectangular or Cartesian form) or the point's polar coordinates (called polar form). The complex number "z" can be represented in rectangular form as where "i" is the imaginary unit, or can alternatively be written in polar form (via the conversion formulae given above) as and from there as where "e" is Euler's number, which are equivalent as shown by Euler's formula. (Note that this formula, like all those involving exponentials of angles, assumes that the angle "φ" is expressed in radians.) To convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used.  For the operations of multiplication, division, and exponentiation of complex numbers, it is generally much simpler to work with complex numbers expressed in polar form rather than rectangular form. From the laws of exponentiation:

Calculus.

Calculus can be applied to equations expressed in polar coordinates.  The angular coordinate "φ" is expressed in radians throughout this section, which is the conventional choice when doing calculus.

Differential calculus.

Using and, one can derive a relationship between derivatives in Cartesian and polar coordinates. For a given function, "u"("x","y"), it follows that (by computing its total derivatives)  or  Hence, we have the following formulae:  Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function "u"("r","φ"), it follows that  or  Hence, we have the following formulae:  To find the Cartesian slope of the tangent line to a polar curve "r"("φ") at any given point, the curve is first expressed as a system of parametric equations.  Differentiating both equations with respect to "φ" yields  Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point :  For other useful formulas including divergence, gradient, and Laplacian in polar coordinates, see curvilinear coordinates.

Integral calculus (arc length).

The arc length (length of a line segment) defined by a polar function is found by the integration over the curve "r"("φ"). Let "L" denote this length along the curve starting from points "A" through to point "B", where these points correspond to "φ" = "a" and "φ" = "b" such that. The length of "L" is given by the following integral

Integral calculus (area).

Let "R" denote the region enclosed by a curve "r"("φ") and the rays "φ" = "a" and "φ" = "b", where. Then, the area of "R" is  This result can be found as follows. First, the interval is divided into "n" subintervals, where "n" is an arbitrary positive integer. Thus Δ"φ", the length of each subinterval, is equal to (the total length of the interval), divided by "n", the number of subintervals. For each subinterval "i" = 1, 2,..., "n", let "φ" be the midpoint of the subinterval, and construct a sector with the center at the pole, radius "r"("φ"), central angle Δ"φ" and arc length "r"("φ")Δ"φ". The area of each constructed sector is therefore equal to Hence, the total area of all of the sectors is  As the number of subintervals "n" is increased, the approximation of the area continues to improve. In the limit as, the sum becomes the Riemann sum for the above integral.  A mechanical device that computes area integrals is the planimeter, which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element linkage effects Green's theorem, converting the quadratic polar integral to a linear integral.

Generalization.

Using Cartesian coordinates, an infinitesimal area element can be calculated as "dA" = "dx" "dy". The substitution rule for multiple integrals states that, when using other coordinates, the Jacobian determinant of the coordinate conversion formula has to be considered:  Hence, an area element in polar coordinates can be written as  Now, a function, that is given in polar coordinates, can be integrated as follows:  Here, "R" is the same region as above, namely, the region enclosed by a curve "r"("φ") and the rays "φ" = "a" and "φ" = "b". The formula for the area of "R" mentioned above is retrieved by taking "f" identically equal to 1.  A more surprising application of this result yields the Gaussian integral, here denoted "K":

Vector calculus.

Vector calculus can also be applied to polar coordinates. For a planar motion, let formula_52 be the position vector, with "r" and "φ" depending on time "t".  We define the unit vectors  in the direction of formula_52 and  in the plane of the motion perpendicular to the radial direction, where formula_56 is a unit vector normal to the plane of the motion.  Then

Centrifugal and Coriolis terms.

The term formula_58 is sometimes referred to as the "centripetal acceleration", and the term formula_59 as the "Coriolis acceleration". For example, see Shankar.  Note: these terms, that appear when acceleration is expressed in polar coordinates, are a mathematical consequence of differentiation; they appear whenever polar coordinates are used. In planar particle dynamics these accelerations appear when setting up Newton's second law of motion in a rotating frame of reference. Here these extra terms are often called fictitious forces; fictitious because they are simply a result of a change in coordinate frame. That does not mean they do not exist, rather they exist only in the rotating frame.

Co-rotating frame.

For a particle in planar motion, one approach to attaching physical significance to these terms is based on the concept of an instantaneous "co-rotating frame of reference". To define a co-rotating frame, first an origin is selected from which the distance "r"("t") to the particle is defined. An axis of rotation is set up that is perpendicular to the plane of motion of the particle, and passing through this origin. Then, at the selected moment "t", the rate of rotation of the co-rotating frame Ω is made to match the rate of rotation of the particle about this axis, "dφ"/"dt". Next, the terms in the acceleration in the inertial frame are related to those in the co-rotating frame. Let the location of the particle in the inertial frame be ("r("t"), "φ"("t")), and in the co-rotating frame be ("r(t), "φ"′(t)"). Because the co-rotating frame rotates at the same rate as the particle, "dφ"′/"dt" = 0. The fictitious centrifugal force in the co-rotating frame is "mrΩ, radially outward. The velocity of the particle in the co-rotating frame also is radially outward, because "dφ"′/"dt" = 0. The "fictitious Coriolis force" therefore has a value −2"m"("dr"/"dt")Ω, pointed in the direction of increasing "φ" only. Thus, using these forces in Newton's second law we find:  where over dots represent time differentiations, and F is the net real force (as opposed to the fictitious forces). In terms of components, this vector equation becomes:  which can be compared to the equations for the inertial frame:  This comparison, plus the recognition that by the definition of the co-rotating frame at time "t" it has a rate of rotation Ω = "dφ"/"dt", shows that we can interpret the terms in the acceleration (multiplied by the mass of the particle) as found in the inertial frame as the negative of the centrifugal and Coriolis forces that would be seen in the instantaneous, non-inertial co-rotating frame.  For general motion of a particle (as opposed to simple circular motion), the centrifugal and Coriolis forces in a particle's frame of reference commonly are referred to the instantaneous osculating circle of its motion, not to a fixed center of polar coordinates. For more detail, see centripetal force.

Differential geometry.

In the modern terminology of differential geometry, polar coordinates provide coordinate charts for the differentiable manifold R \ {(0,0)}, the plane minus the origin. In these coordinates, the Euclidean metric tensor is given byformula_63This can be seen via the change of variables formula for the metric tensor, or by computing the differential forms "dx", "dy" via the exterior derivative of the 0-forms "x = r cos(θ), y = r sin(θ)" and substituting them in the Euclidean metric tensor "ds = dx + dy". An orthonormal frame with respect to this metric is given byformula_64with dual coframeformula_65The connection form relative to this frame and the Levi-Civita connection is given by the skew-symmetric matrix of 1-formsformula_66and hence the curvature form Ω = dω + ω∧ω vanishes identically. Therefore, as expected, the punctured plane is a flat manifold.

Extensions in 3D.

The polar coordinate system is extended into three dimensions with two different coordinate systems, the cylindrical and spherical coordinate system.

Applications.

Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane. They are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point. For instance, the examples above show how elementary polar equations suffice to define curves—such as the Archimedean spiral—whose equation in the Cartesian coordinate system would be much more intricate. Moreover, many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates. The initial motivation for the introduction of the polar system was the study of circular and orbital motion.

Position and navigation.

Polar coordinates are used often in navigation as the destination or direction of travel can be given as an angle and distance from the object being considered. For instance, aircraft use a slightly modified version of the polar coordinates for navigation. In this system, the one generally used for any sort of navigation, the 0° ray is generally called heading 360, and the angles continue in a clockwise direction, rather than counterclockwise, as in the mathematical system. Heading 360 corresponds to magnetic north, while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively. Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read zero-niner-zero by air traffic control).

Modeling.

Systems displaying radial symmetry provide natural settings for the polar coordinate system, with the central point acting as the pole. A prime example of this usage is the groundwater flow equation when applied to radially symmetric wells. Systems with a radial force are also good candidates for the use of the polar coordinate system. These systems include gravitational fields, which obey the inverse-square law, as well as systems with point sources, such as radio antennas.  Radially asymmetric systems may also be modeled with polar coordinates. For example, a microphone's pickup pattern illustrates its proportional response to an incoming sound from a given direction, and these patterns can be represented as polar curves. The curve for a standard cardioid microphone, the most common unidirectional microphone, can be represented as at its target design frequency. The pattern shifts toward omnidirectionality at lower frequencies.

References.

Specific General

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system) is called the "pole", and the ray from the pole in the reference direction is the "polar axis". The distance from the pole is called the "radial coordinate", "radial distance" or simply "radius", and the angle is called the "angular coordinate", "polar angle", or "azimuth". The radial coordinate is often denoted by "r" or "ρ", and the angular coordinate by "φ", "θ", or "t". Angles in polar notation are generally expressed in either degrees or radians (2 rad being equal to 360°).

Polární soustava souřadnic je taková soustava souřadnic v rovině, u které jedna souřadnice (označovaná formula_1) udává vzdálenost bodu od počátku souřadnic, druhá souřadnice (označovaná formula_2) udává úhel spojnice tohoto bodu a počátku od zvolené osy ležící v rovině (nejčastěji jí odpovídá osa formula_3 kartézských souřadnic).


Definitions.

For cyclical processes, such as rotation, oscillations, or waves, frequency is defined as a number of cycles per unit time. In physics and engineering disciplines, such as optics, acoustics, and radio, frequency is usually denoted by a Latin letter "f" or by the Greek letter "formula_1" or "ν" (nu) (see e.g. Planck's formula).  The relation between the frequency and the period, formula_2, of a repeating event or oscillation is given by

Units.

The SI derived unit of frequency is the hertz (Hz), named after the German physicist Heinrich Hertz. One hertz means that an event repeats once per second. If a TV has a refresh rate of 1 hertz the TV's screen will change (or refresh) its picture once per second. A previous name for this unit was cycles per second (cps). The SI unit for period is the second.  A traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated r/min or rpm. 60 rpm equals one hertz.

Period versus frequency.

As a matter of convenience, longer and slower waves, such as ocean surface waves, tend to be described by wave period rather than frequency. Short and fast waves, like audio and radio, are usually described by their frequency instead of period. These commonly used conversions are listed below:

In wave propagation.

For periodic waves in nondispersive media (that is, media in which the wave speed is independent of frequency), frequency has an inverse relationship to the wavelength, "λ" (lambda). Even in dispersive media, the frequency "f" of a sinusoidal wave is equal to the phase velocity "v" of the wave divided by the wavelength "λ" of the wave:  In the special case of electromagnetic waves moving through a vacuum, then "v = c", where "c" is the speed of light in a vacuum, and this expression becomes:  When waves from a monochrome source travel from one medium to another, their frequency remains the same—only their wavelength and speed change.

Measurement.

Measurement of frequency can be done in the following ways,

Counting.

Calculating the frequency of a repeating event is accomplished by counting the number of times that event occurs within a specific time period, then dividing the count by the length of the time period. For example, if 71 events occur within 15 seconds the frequency is: If the number of counts is not very large, it is more accurate to measure the time interval for a predetermined number of occurrences, rather than the number of occurrences within a specified time. The latter method introduces a random error into the count of between zero and one count, so on average half a count. This is called "gating error" and causes an average error in the calculated frequency of formula_11, or a fractional error of formula_12 where formula_2 is the timing interval and formula_14 is the measured frequency. This error decreases with frequency, so it is generally a problem at low frequencies where the number of counts N is small.

Stroboscope.

An older method of measuring the frequency of rotating or vibrating objects is to use a stroboscope. This is an intense repetitively flashing light (strobe light) whose frequency can be adjusted with a calibrated timing circuit. The strobe light is pointed at the rotating object and the frequency adjusted up and down. When the frequency of the strobe equals the frequency of the rotating or vibrating object, the object completes one cycle of oscillation and returns to its original position between the flashes of light, so when illuminated by the strobe the object appears stationary. Then the frequency can be read from the calibrated readout on the stroboscope. A downside of this method is that an object rotating at an integral multiple of the strobing frequency will also appear stationary.

Frequency counter.

Higher frequencies are usually measured with a frequency counter. This is an electronic instrument which measures the frequency of an applied repetitive electronic signal and displays the result in hertz on a digital display. It uses digital logic to count the number of cycles during a time interval established by a precision quartz time base. Cyclic processes that are not electrical in nature, such as the rotation rate of a shaft, mechanical vibrations, or sound waves, can be converted to a repetitive electronic signal by transducers and the signal applied to a frequency counter. As of 2018, frequency counters can cover the range up to about 100 GHz. This represents the limit of direct counting methods; frequencies above this must be measured by indirect methods.

Heterodyne methods.

Above the range of frequency counters, frequencies of electromagnetic signals are often measured indirectly by means of heterodyning (frequency conversion). A reference signal of a known frequency near the unknown frequency is mixed with the unknown frequency in a nonlinear mixing device such as a diode. This creates a heterodyne or "beat" signal at the difference between the two frequencies. If the two signals are close together in frequency the heterodyne is low enough to be measured by a frequency counter. This process only measures the difference between the unknown frequency and the reference frequency. To reach higher frequencies, several stages of heterodyning can be used. Current research is extending this method to infrared and light frequencies (optical heterodyne detection).

Examples.

Light.

Visible light is an electromagnetic wave, consisting of oscillating electric and magnetic fields traveling through space. The frequency of the wave determines its color: is red light, is violet light, and between these (in the range 4-) are all the other colors of the visible spectrum. An electromagnetic wave can have a frequency less than, but it will be invisible to the human eye; such waves are called infrared (IR) radiation. At even lower frequency, the wave is called a microwave, and at still lower frequencies it is called a radio wave. Likewise, an electromagnetic wave can have a frequency higher than, but it will be invisible to the human eye; such waves are called ultraviolet (UV) radiation. Even higher-frequency waves are called X-rays, and higher still are gamma rays.  All of these waves, from the lowest-frequency radio waves to the highest-frequency gamma rays, are fundamentally the same, and they are all called electromagnetic radiation. They all travel through a vacuum at the same speed (the speed of light), giving them wavelengths inversely proportional to their frequencies.  where "c" is the speed of light ("c" in a vacuum, or less in other media), "f" is the frequency and λ is the wavelength.  In dispersive media, such as glass, the speed depends somewhat on frequency, so the wavelength is not quite inversely proportional to frequency.

Sound.

Sound propagates as mechanical vibration waves of pressure and displacement, in air or other substances.. In general, frequency components of a sound determine its "color", its timbre. When speaking about the frequency (in singular) of a sound, it means the property that most determines pitch.  The frequencies an ear can hear are limited to a specific range of frequencies. The audible frequency range for humans is typically given as being between about 20 Hz and 20,000 Hz (20 kHz), though the high frequency limit usually reduces with age. Other species have different hearing ranges. For example, some dog breeds can perceive vibrations up to 60,000 Hz.  In many media, such as air, the speed of sound is approximately independent of frequency, so the wavelength of the sound waves (distance between repetitions) is approximately inversely proportional to frequency.

Line current.

In Europe, Africa, Australia, Southern South America, most of Asia, and Russia, the frequency of the alternating current in household electrical outlets is 50 Hz (close to the tone G), whereas in North America and Northern South America, the frequency of the alternating current in household electrical outlets is 60 Hz (between the tones B♭ and B; that is, a minor third above the European frequency). The frequency of the 'hum' in an audio recording can show where the recording was made, in countries using a European, or an American, grid frequency.

Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. Frequency is measured in units of hertz (Hz) which is equal to one occurrence of a repeating event per second. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute (2 hertz), its period,, — the time interval between beats—is half a second (60 seconds divided by 120 beats). Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.

Frekvence (též kmitočet) je fyzikální veličina, která udává počet opakování periodického děje za daný časový úsek. Například v obvodu střídavého proudu takto označujeme počet kmitů napětí či proudu za jednotku času.


Systematics, taxonomy, and evolution.

The name Corvidae for the family was introduced by the English zoologist William Elford Leach in a guide to the contents of the British Museum published in 1820. Over the years, much disagreement has arisen on the exact evolutionary relationships of the corvid family and their relatives. What eventually seemed clear was that corvids are derived from Australasian ancestors and from there spread throughout the world. Other lineages derived from these ancestors evolved into ecologically diverse, but often Australasian groups. In the late 1970s and throughout the 1980s, Sibley and Ahlquist united the corvids with other taxa in the Corvida, based on DNA–DNA hybridization. The presumed corvid relatives included currawongs, birds of paradise, whipbirds, quail-thrushes, whistlers, monarch flycatchers and drongos, shrikes, vireos, and vangas, but current research favors the theory that this grouping is partly artificial. The corvids constitute the core group of the Corvoidea, together with their closest relatives (the birds of paradise, Australian mud-nesters, and shrikes). They are also the core group of the Corvida, which includes the related groups, such as Old World orioles and vireos.  Clarification of the interrelationships of the corvids has been achieved based on cladistic analysis of several DNA sequences. The jays and magpies do not constitute monophyletic lineages, but rather seem to split up into an American and Old World lineage, and an Holarctic and Oriental lineage, respectively. These are not closely related among each other. The position of the azure-winged magpie, which has always been a major enigma, is even less clear than before.  The crested jay ("Platylophus galericulatus") is traditionally included in the Corvidae, but might not be a true member of this family, possibly being closer to the helmetshrikes (Malaconotidae) or shrikes (Laniidae); it is best considered Corvidae "incertae sedis" for the time being. Likewise, the Hume's ground "jay" ("Pseudopodoces humilis") is in fact a member of the tit family Paridae. The following tree represents current insights in the phylogeny of the Crow family according to J. Boyd.

Fossil record.

The earliest corvid fossils date to mid-Miocene Europe, about 17 million years ago; "Miocorvus" and "Miopica" may be ancestral to crows and some of the magpie lineage, respectively, or similar to the living forms due to convergent evolution. The known prehistoric corvid genera appear to be mainly of the New World and Old World jay and Holarctic magpie lineages:  In addition, there are numerous fossil species of extant genera since the Mio–Pliocene, mainly European "Corvus".

Morphology.

Corvids are large to very large passerines with a robust build, strong legs and all species except the pinyon jay have nostrils covered by bristle-like feathers. Many corvids of temperate zones have mainly black or blue coloured plumage; however, some are pied black and white, some have a blue-purple iridescence and many tropical species are brightly coloured. The sexes are very similar in color and size. Corvids have strong, stout bills and large wingspans. The family includes the largest members of the passerine order.  The smallest corvid is the dwarf jay ("Aphelocoma nana"), at and. The largest corvids are the common raven ("Corvus corax") and the thick-billed raven ("Corvus crassirostris"), both of which regularly exceed and.  Species can be identified based on size, shape, and geography; however, some, especially the Australian crows, are best identified by their raucous calls.

Ecology.

Corvids occur in most climatic zones. Most are sedentary and do not migrate significantly. However, during a shortage of food, irruptive migration can occur. When species are migratory, they will form large flocks in the fall (around August in the Northern Hemisphere) and travel south.  One reason for the success of crows, compared to ravens, is their ability to overlap breeding territory. During breeding season, crows were shown to overlap breeding territory six times as much as ravens. This invasion of breeding ranges allowed a related increase in local population density.  Since crows and magpies have benefited and even increased in numbers due to human development, it was suggested that this might cause increased rates of nest predation of smaller bird species, leading to declines. Several studies have shown this concern to be unfounded. One study examined American crows, which had increased in numbers, were a suspect in nest predation of threatened marbled murrelets. However, Steller's jays, which are successful independently of human development, are more efficient in plundering small birds' nests than American crows and common ravens. Therefore, the human relationship with crows and ravens did not significantly increase nest predation, compared to other factors such as habitat destruction. Similarly a study examining the decline of British songbirds found no link between Eurasian magpie numbers and population changes of 23 songbird species.

Behaviour.

Some corvids have strong organization and community groups. Jackdaws, for example, have a strong social hierarchy, and are facultatively colonial during breeding. Providing mutual aid has also been recorded within many of the corvid species.  Young corvids have been known to play and take part in elaborate social games. Documented group games follow "king of the mountain", or "follow the leader", patterns. Other play involves the manipulation, passing, and balancing of sticks. Corvids also take part in other activities, such as sliding down smooth surfaces. These games are understood to play a large role in the adaptive and survival ability of the birds.  Mate selection is quite complex and accompanied with much social play in the Corvidae. Youngsters of social corvid species undergo a series of tests, including aerobatic feats, before being accepted as a mate by the opposite sex.  Some corvids can be aggressive. Blue jays, for example, are well known to attack anything that threatens their nest. Crows have been known to attack dogs, cats, ravens, and birds of prey. Most of the time these assaults take place as a distraction long enough to allow an opportunity for stealing food.

Food and feeding.

The natural diet of many corvid species is omnivorous, consisting of invertebrates, nestlings, small mammals, berries, fruits, seeds, and carrion. However, some corvids, especially the crows, have adapted well to human conditions and have come to rely on anthropogenic foods. In a US study of American crows, common ravens and Steller's jays around campgrounds and human settlements, the crows appeared to have the most diverse diet of all, taking anthropogenic foods such as bread, spaghetti, fried potatoes, dog food, sandwiches, and livestock feed. The increase in available anthropogenic food sources is contributing to population increase in some corvid species.  Some corvids are predators of other birds. During the wintering months, corvids typically form foraging flocks. However, some crows also eat many agricultural pests including cutworms, wireworms, grasshoppers, and harmful weeds. Some corvids will eat carrion, and since they lack a specialized beak for tearing into flesh, they must wait until animals are opened, whether by other predators or as roadkill.

Reproduction.

Many species of corvid are territorial, protecting territories throughout the year or simply during the breeding season. In some cases territories may only be guarded during the day, with the pair joining off-territory roosts at night. Some corvids are well-known communal roosters. Some groups of roosting corvids can be very large, with a roost of 65,000 rooks counted in Scotland. Some, including the rook and the jackdaw, are also communal nesters.  The partner bond in corvids is extremely strong and even lifelong in some species. This monogamous lifestyle, however, can still contain extra-pair copulations. Males and females build large nests together in trees or on ledges; jackdaws are known to breed in buildings and in rabbit warrens. The male will also feed the female during incubation. The nests are constructed of a mass of bulky twigs lined with grass and bark. Corvids can lay between 3 and 10 eggs, typically ranging between 4 and 7. The eggs are usually greenish in colour with brown blotches. Once hatched, the young remain in the nests for up to 6–10 weeks depending on the species.  Corvids use several different forms of parental care, including biparental care and cooperative breeding. Cooperative breeding takes place when parents are helped in raising their offspring, usually by relatives but also sometimes by non-related adults. Such helpers at the nest in most cooperatively breeding birds are males, while females join other groups.White-throated magpie-jays are cooperatively breeding corvids where the helpers are mostly female.

Intelligence.

The brain-to-body weight ratios of corvid brains are among the largest in birds, equal to that of most great apes and cetaceans, and only slightly lower than a human. Their intelligence is boosted by the long growing period of the young. By remaining with the parents, the young have more opportunities to learn necessary skills.  When compared to dogs and cats in an experiment testing the ability to seek out food according to three-dimensional clues, corvids out-performed the mammals. A meta-analysis testing how often birds invented new ways to acquire food in the wild found corvids to be the most innovative birds. A 2004 review suggests that their cognitive abilities are on par with those of great apes. Despite structural differences, the brains of corvids and great apes both evolved the ability to make geometrical measurements.  Corvid ingenuity is represented through their feeding skills, memorization abilities, use of tools, and group behaviour. Living in large social groups has long been connected with high cognitive ability. To live in a large group, a member must be able to recognize individuals and track the social position and foraging of other members over time. Members must also be able to distinguish between sex, age, reproductive status, and dominance, and to update this information constantly. It might be that social complexity corresponds to their high cognition.  The Eurasian magpie is the only non-mammal species known to be able to recognize itself in a mirror test. Magpies have been observed taking part in elaborate grieving rituals, which have been likened to human funerals, including laying grass wreaths. Marc Bekoff, at the University of Colorado, argues that it shows that they are capable of feeling complex emotions, including grief.  There are also specific examples of corvid cleverness. One carrion crow was documented to crack nuts by placing them on a crosswalk, letting the passing cars crack the shell, waiting for the light to turn red, and then safely retrieving the contents. A group of crows in England took turns lifting garbage bin lids while their companions collected food.  Members of the corvid family have been known to watch other birds, remember where they hide their food, then return once the owner leaves. Corvids also move their food around between hiding places to avoid thievery, but only if they have previously been thieves themselves (that is, they remember previous relevant social contexts, use their own experience of having been a thief to predict the behavior of a pilferer, and can determine the safest course to protect their caches from being pilfered). Studies to assess similar cognitive abilities in apes have been inconclusive.  The ability to hide food requires highly accurate spatial memories. Corvids have been recorded to recall their food's hiding place up to nine months later. It is suggested that vertical landmarks (like trees) are used to remember locations. There has also been evidence that California scrub jays, which store perishable foods, not only remember where they stored their food, but for how long. This has been compared to episodic memory, previously thought unique to humans.  New Caledonian crows ("Corvus moneduloides") are notable for their highly developed tool fabrication. They make angling tools of twigs and leaves trimmed into hooks, then use the hooks to pull insect larvae from tree holes. Tools are engineered according to task and apparently also to learned preference. Recent studies revealed abilities to solve complicated problems, which suggests high level of innovation of a complex nature. Other corvids that have been observed using tools include the American crow, blue jay and green jay. Researchers have discovered that New Caledonian crows don't just use single objects as tools; they can also construct novel compound tools through assemblage of otherwise non-functional elements. Diversity in tool design among corvids suggests cultural variation. Again, great apes are the only other animals known to use tools in such a fashion.  Clark's nutcrackers and jackdaws were compared in a 2002 study based on geometric rule learning. The corvids, along with a domestic pigeon, had to locate a target between two landmarks, while distances and landmarks were altered. The nutcrackers were more accurate in their searches than the jackdaws and pigeons.  The scarecrow is an archetypal scare tactic in the agricultural business. However, due to corvids' quick wit, scarecrows are soon ignored and used as perches. Despite farmers' efforts to rid themselves of corvid pests, their attempts have only expanded corvid territories and strengthened their numbers.  Contrary to earlier teleological classifications in which they were seen as "highest" songbirds due to their intelligence, current systematics might place corvids, based on their total number of physical characteristics instead of just their brains (which are the most developed of birds), in the lower middle of the passerine evolutionary tree, dependent on which subgroup is chosen as the most derived. As per one observer:  The other major group of highly intelligent birds of the order Psittaciformes (which includes 'true' parrots, cockatoos and New Zealand parrots) is not closely related to corvids.

Disease.

Corvids are reservoirs (carriers) for the West Nile virus in the United States. They are infected by mosquitoes (the vectors), primarily of the "Culex" species. Crows and ravens are quickly killed by this disease, so their deaths are an early-warning system when West Nile virus arrives in an area (as are horse and other bird species deaths). One of the first signs that West Nile virus first arrived in the US in 1999 was the death of crows in New York.

Relationship with humans.

Several different corvids, particularly ravens, have occasionally served as pets, although they are not able to speak as readily as parrots and do not like being caged.

Role in myth and culture.

Folklore often represents corvids as clever, and even mystical, animals. Some Native Americans, such as the Haida, believed that a raven created the earth and despite being a trickster spirit, ravens were popular on totems, credited with creating man, and considered responsible for placing the Sun in the sky.  Due to their carrion diet, the Celtic peoples strongly associated corvids with war, death and the battlefield – their great intelligence meant that they were often considered messengers, or manifestations of the gods such Bendigeidfran "Blessed raven" or the Irish Morrigan, underworld deities that may be related to the later Arthurian Fisher King. The Welsh "Dream of Rhonabwy" illustrates well the association of ravens with war. In many parts of Britain, gatherings of crows, or more often magpies, are counted using the divination rhyme: "one for sorrow, two for joy, three for a girl, four for a boy, five for silver, six for gold, seven for a secret never to be told." Another rhyme is: "one for sorrow, two for mirth, three for a funeral, four for a birth, five for heaven, six for hell, and seven for the Devil, his own sel'." Cornish superstition holds that when a lone magpie is encountered, it must be loudly greeted with respect.  Various Germanic peoples highly revered the raven. The major deity Odin was so associated with ravens throughout history that he gained the kenning "raven god" and the raven banner was the flag of various Viking Age Scandinavian chieftains. He was also attended by Hugin and Munin, two ravens who whispered news into his ears. The Valravn sometimes appears in modern Scandinavian folklore. The Sutton Hoo treasure features stylised corvids with scrolled beaks in the decorative enamel work on the shield and purse lid reflecting their common totemic status to the Anglo-Saxons, whose pre-Christian indigenous beliefs were of the same origin as that of the aforementioned Vikings.  The 6th century BC Greek scribe Aesop featured corvids as intelligent antagonists in many fables. Later, in western literature, popularized by American poet Edgar Allan Poe's work "The Raven", the common raven becomes a symbol of the main character's descent into madness.  The book "Mrs. Frisby and the Rats of NIMH" and its film adaptation features a crow named Jeremy.

Status and conservation.

Unlike many other bird families, corvid fitness and reproduction, especially with many crows, has increased due to human development. The survival and reproductive success of certain crows and ravens is assisted by their close relationship with humans.  Human development provides additional resources by clearing land, creating shrublands rich in berries and insects. When the cleared land naturally replenishes, jays and crows use the young dense trees for nesting sites. Ravens typically use larger trees in denser forest.  Despite the fact that most corvids are not threatened (many even increasing due to human activity) a few species are in danger. For example, the destruction of the Southeast Asian rainforest is endangering mixed-species feeding flocks with members from the family Corvidae. Also, since its semiarid scrubland habitat is an endangered ecosystem, the Florida scrub jay has a small and declining population. A number of island species, which are more vulnerable to introduced species and habitat loss, have been driven to extinction, such as the New Zealand raven, or are threatened, like the Mariana crow.  The American crow population of the United States has grown over the years. It is possible that the American crow, due to humans increasing suitable habitat, will cause Northwestern crows and fish crows to decline.

Species.

FAMILY CORVIDAE

Corvidae is a cosmopolitan family of oscine passerine birds that contains the crows, ravens, rooks, jackdaws, jays, magpies, treepies, choughs, and nutcrackers. In common English, they are known as the crow family, or, more technically, corvids. Over 120 species are described. The genus "Corvus", including the jackdaws, crows, rooks, and ravens, makes up over a third of the entire family. Corvids are the largest passerines.

Krkavcovití (Corvidae) jsou největší čeledí řádu pěvců. Patří mezi všežravce, hnízdí na stromech, skalách nebo ve skalních dutinách. V Česku se vyskytujícími známými druhy jsou například vrána černá a vrána šedá, havran polní, krkavec velký, kavka obecná, sojka obecná, straka obecná, ořešník kropenatý.


History.

Italian scientist Giovanni Battista Riccioli and his assistant Francesco Maria Grimaldi described the effect in connection with artillery in the 1651 "Almagestum Novum", writing that rotation of the Earth should cause a cannonball fired to the north to deflect to the east. In 1674 Claude François Milliet Dechales described in his "Cursus seu Mundus Mathematicus" how the rotation of the Earth should cause a deflection in the trajectories of both falling bodies and projectiles aimed toward one of the planet's poles. Riccioli, Grimaldi, and Dechales all described the effect as part of an argument against the heliocentric system of Copernicus. In other words, they argued that the Earth's rotation should create the effect, and so failure to detect the effect was evidence for an immobile Earth. The Coriolis acceleration equation was derived by Euler in 1749, and the effect was described in

Formula.

In Newtonian mechanics, the equation of motion for an object in an inertial reference frame is where formula_2 is the vector sum of the physical forces acting on the object, formula_3 is the mass of the object, and formula_4 is the acceleration of the object relative to the inertial reference frame.  Transforming this equation to a reference frame rotating about a fixed axis through the origin with rotation vector formula_5 having variable rotation rate, the equation takes the form  where  The fictitious forces as

Causes.

The Coriolis force exists only when one uses a rotating reference frame. In the rotating frame it behaves exactly like a real force (that is to say, it causes acceleration and has real effects). However, the Coriolis force is a consequence of inertia, and is not attributable to an identifiable originating body, as is the case for electromagnetic or nuclear forces, for example. From an analytical viewpoint, to use Newton's second law in a rotating system, the Coriolis force is mathematically necessary, but it disappears in a non-accelerating, inertial frame of reference. For example, consider two children on opposite sides of a spinning roundabout (Merry-go-round), who are throwing a ball to each other. From the children's point of view, this ball's path is curved sideways by the Coriolis force. Suppose the roundabout spins anticlockwise when viewed from above. From the thrower's perspective, the deflection is to the right. From the non-thrower's perspective, deflection is to the left (for a mathematical formulation see Mathematical derivation of fictitious forces). In meteorology, a rotating frame (the Earth) with its Coriolis force provides a more natural framework for explanation of air movements than a non-rotating inertial frame without Coriolis

Length scales and the Rossby number.

The time, space and velocity scales are important in determining the importance of the Coriolis force. Whether rotation is important in a system can be determined by its Rossby number, which is the ratio of the velocity, "U", of a system to the product of the Coriolis parameter,formula_23, and the length scale, "L", of the motion: The Rossby number is the ratio of inertial to Coriolis forces. A small Rossby number indicates a system is strongly affected by Coriolis forces, and a large Rossby number indicates a system in which inertial forces dominate. For example, in tornadoes, the Rossby number is large, in low-pressure systems it is low, and in oceanic systems it is around 1. As a

Simple cases.

Cannon on turntable.

The animation at the top of this article is a classic illustration of Coriolis force. Another visualization of the Coriolis and centrifugal forces is this animation clip.  Given the radius "R" of the turntable in that animation, the rate of angular rotation ω, and the speed of the cannonball (assumed constant) "v", the correct angle θ to aim so as to hit the target at the edge of the turntable can be calculated.  The inertial frame of reference provides one way to handle the question: calculate the time to interception, which is "t" = "R" / "v". Then, the turntable revolves an angle ω "t" in this time. If the cannon is pointed an angle θ = ω "t" = ω "R" / "v", then the cannonball arrives at the periphery at position number 3 at the same time as the target.  No discussion of Coriolis force can arrive at this solution as simply, so the reason to treat this problem is to demonstrate Coriolis formalism in an easily visualized situation.

Trajectory in the inertial frame.

The trajectory in the inertial frame (denoted "A") is a straight line radial path at angle θ. The position of the cannonball in ("x", "y") coordinates at time "t" is: In the turntable frame (denoted "B"), the "x"- "y" axes rotate at angular rate ω, so the trajectory becomes: and three examples of this result are plotted in the figure.

Tossed ball on a rotating carousel.

The figure illustrates a ball tossed from 12:00 o'clock toward the center of a counter-clockwise rotating carousel. On the left, the ball is seen by a stationary observer above the carousel, and the ball travels in a straight line to the center, while the ball-thrower rotates counter-clockwise with the carousel. On the right the ball is seen by an observer rotating with the carousel, so the ball-thrower appears to stay at 12:00 o'clock. The figure shows how the trajectory of the ball as seen by the rotating observer can be constructed.  On the left, two arrows locate the ball relative to the ball-thrower. One of these arrows is from the thrower to the center of the carousel (providing the ball-thrower's line of sight), and the other points from the center of the carousel to the ball. (This arrow gets

Bounced ball.

The figure describes a more complex situation where the tossed ball on a turntable bounces off the edge of the carousel and then returns to the tosser, who catches the ball. The effect of Coriolis force on its trajectory is shown again as seen by two observers: an observer (referred to as the "camera") that rotates with the carousel, and an inertial observer. The figure shows a bird's-eye view based upon the same ball speed on forward and return paths. Within each circle, plotted dots show the same time points. In the left panel, from the camera's viewpoint at the center of rotation, the tosser (smiley face) and the rail both are at fixed locations, and the ball makes a very considerable arc on its travel toward the rail, and takes a more direct route on the way back. From the ball tosser's viewpoint, the ball seems to return more quickly than it went (because the tosser is rotating toward the ball on the return flight).  On the carousel, instead of

Applied to the Earth.

The concept "Coriolis force" is specially suitable for the description of motion of atmosphere (i.e. winds) over the surface of the Earth. The Earth (like all rotating celestial bodies) has taken the shape of an oblate spheroid, such that the gravitational force is slightly off-set towards the Earth axis as illustrated in the figure. For a mass point at rest on the Earth surface the horizontal component of the gravitation counteracts the "centrifugal force" preventing it to slide away towards the equator. This means that the vector term of the "equation of motion" above  is directed straight down, orthogonal to the surface of the Earth. The force affecting the motion of air "sliding" over the Earth surface is therefore (only) the horizontal component of the Coriolis term  This component is orthogonal to the velocity over the Earth surface and is given by the expression where  In the northern hemisphere where the sign is positive this force/acceleration, as viewed from above, is to the right of the direction of motion, in the southern hemisphere where the sign is negative this force/acceleration is to the left of the direction of motion

Intuitive explanation.

As the Earth rotates about its axis, everything attached to it, including the atmosphere, turns with it (imperceptibly to our senses). An object that is moving without being dragged along with the surface rotation or atmosphere such as an object in ballistic flight or an independent air mass within the atmosphere, travels in a straight motion over the turning Earth. From our rotating perspective on the planet, the direction of motion

Rotating sphere.

Consider a location with latitude "φ" on a sphere that is rotating around the north-south axis. A local coordinate system is set up with the "x" axis horizontally due east, the "y" axis horizontally due north and the "z" axis vertically upwards. The rotation vector, velocity of movement and Coriolis acceleration expressed in this local coordinate system (listing components in the order east ("e"), north ("n") and upward ("u")) are:  When considering atmospheric or oceanic dynamics, the vertical velocity is small, and the vertical component of the Coriolis acceleration is small compared with the acceleration due to gravity. For such cases, only the horizontal (east and north)

Meteorology.

Perhaps the most important impact of the Coriolis effect is in the large-scale dynamics of the oceans and the atmosphere. In meteorology and oceanography, it is convenient to postulate a rotating frame of reference wherein the Earth is stationary. In accommodation of that provisional postulation, the centrifugal and Coriolis forces are introduced. Their relative importance is determined by the applicable Rossby numbers. Tornadoes have high Rossby numbers, so, while tornado-associated centrifugal forces are quite substantial, Coriolis forces associated with tornadoes are for practical purposes negligible.  Because surface ocean currents are driven by the movement of wind over the water's surface, the Coriolis force

Flow around a low-pressure area.

If a low-pressure area forms in the atmosphere, air tends to flow in towards it, but is deflected perpendicular to its velocity by the Coriolis force. A system of equilibrium can then establish itself creating circular movement, or a cyclonic flow. Because the Rossby number is low, the force balance is largely between the pressure-gradient force acting towards the low-pressure area and the Coriolis force acting away from the center of the low pressure.  Instead of flowing down the gradient, large scale motions in the atmosphere and ocean tend to occur perpendicular to

Inertial circles.

An air or water mass moving with speed formula_48 subject only to the Coriolis force travels in a circular trajectory called an 'inertial circle'. Since the force is directed at right angles to the motion of the particle, it moves with a constant speed around a circle whose radius formula_49 is given by:  where formula_51 is the Coriolis parameter formula_52, introduced above (where formula_53 is the latitude). The time taken for the mass to complete a full circle is therefore formula_54. The Coriolis parameter typically has a mid-latitude

Other terrestrial effects.

The Coriolis effect strongly affects the large-scale oceanic and atmospheric circulation, leading to the formation of robust features like jet streams and western boundary currents. Such features are in geostrophic balance, meaning that the Coriolis and "pressure gradient" forces balance each other. Coriolis acceleration is also responsible for the propagation of many types of waves in the ocean and atmosphere, including Rossby waves and Kelvin waves. It is also instrumental in the so-called Ekman dynamics in the ocean, and in the establishment of the large-scale ocean flow pattern called the Sverdrup balance.

Eötvös effect.

The practical impact of the "Coriolis effect" is mostly caused by the horizontal acceleration component produced by horizontal motion.  There are other components of the Coriolis effect. Westward-travelling objects are deflected downwards (feel heavier), while Eastward-travelling objects are deflected upwards (feel lighter). This is known as the Eötvös effect. This aspect of the Coriolis effect is greatest near the equator. The force produced by the Eötvös effect is similar to the horizontal component, but the much larger vertical forces due to gravity and pressure suggest that it is unimportant in the hydrostatic equilibrium. However, in the atmosphere, winds are associated with small deviations of pressure

Intuitive example.

Imagine a train that travels through a frictionless railway line along the equator. Assume that, when in motion, it moves at the necessary speed to complete a trip around the world in one day (465 m/s). The Coriolis effect can be considered in three cases: when the train travels west, when it is at rest, and when it travels east. In each case, the Coriolis effect can be calculated from the rotating frame of reference on Earth first, and then checked against a fixed inertial frame. The image below illustrates the three cases as viewed by an observer at rest in a (near) inertial frame from a fixed point above the North Pole along the Earth's axis of rotation; the train is denoted by a few red pixels, fixed at the left side in the leftmost picture, moving in the others formula_58  This also explains why high speed projectiles that travel west are deflected down, and those that travel east are deflected up. This vertical component

Draining in bathtubs and toilets.

Contrary to popular misconception, bathtubs, toilets and other residential water receptacles do not drain in opposite directions in the Northern and Southern Hemispheres because the magnitude of the Coriolis force is negligible at this scale. Forces determined by the initial conditions of the water (e.g. the geometry of the drain, the geometry of the receptacle, pre-existing momentum of water, etc.) are likely to be orders of magnitude greater than the Coriolis force and hence will determine the direction of water rotation, if any. For example, identical toilets flushed in each hemisphere will drain in the same direction, and this direction

Ballistic trajectories.

The Coriolis force is important in external ballistics for calculating the trajectories of very long-range artillery shells. The most famous historical example was the Paris gun, used by the Germans during World War I to bombard Paris from a range of about. The Coriolis force minutely changes the trajectory of a bullet, affecting accuracy at extremely long distances. It is adjusted for by accurate long-distance shooters, such as snipers. At the latitude of Sacramento a 1000-yard shot

Visualization of the Coriolis effect.

To demonstrate the Coriolis effect, a parabolic turntable can be used. On a flat turntable, the inertia of a co-rotating object forces it off the edge. However, if the turntable surface has the correct paraboloid (parabolic bowl) shape (see the figure) and rotates at the corresponding rate, the force components shown in the figure make the component of gravity tangential to the bowl surface exactly equal to the centripetal force necessary to keep the object rotating at its velocity and radius of curvature (assuming no friction). (See.) This carefully contoured surface allows the Coriolis force to be displayed in isolation.  Discs cut from cylinders of dry ice can be used as pucks, moving around almost frictionlessly over the surface of the parabolic turntable, allowing effects of Coriolis on dynamic phenomena to show themselves. To get a view of the motions as seen from the reference frame rotating with the turntable, a video camera is attached to the turntable so as to co-rotate with the turntable, with results as shown in the figure. In the left panel

Coriolis effects in other areas.

Coriolis flow meter.

A practical application of the Coriolis effect is the mass flow meter, an instrument that measures the mass flow rate and density of a fluid flowing through a tube. The operating principle involves inducing a vibration of the tube through which

Molecular physics.

In polyatomic molecules, the molecule motion can be described by a rigid body rotation and internal vibration of atoms about their equilibrium position. As a result of the vibrations of the atoms, the atoms

Gyroscopic precession.

When an external torque is applied to a spinning gyroscope along an axis that is at right angles to the spin axis, the rim velocity that is associated with the spin becomes radially directed in

Insect flight.

Flies (Diptera) and some moths (Lepidoptera) exploit the Coriolis effect in flight with specialized appendages and organs that relay information about the angular velocity of their bodies.  Coriolis forces resulting from linear motion of these appendages are detected within the rotating frame of reference of the insects' bodies. In the case of flies, their specialized appendages are dumbbell shaped organs located just behind their wings called "halteres".

Lagrangian point stability.

In astronomy, Lagrangian points are five positions in the orbital plane of two large orbiting bodies where a small object affected only by gravity can maintain a stable position relative to the two large bodies. The first three Lagrangian points (L, L, L) lie along the line connecting the two large bodies, while the last

In physics, the Coriolis force is an inertial or fictitious force that acts on objects that are in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the left of the motion of the object. In one with anticlockwise (or counterclockwise) rotation, the force acts to the right. Deflection of an object due to the Coriolis force is called the Coriolis effect. Though recognized previously by others, the mathematical expression for the Coriolis force appeared in an 1835 paper by French scientist Gaspard-Gustave de Coriolis, in connection with the theory of water wheels. Early in the 20th century, the term "Coriolis force" began to be used in connection with meteorology.

Coriolisova síla je setrvačná síla působící na tělesa, která se pohybují v rotující neinerciální vztažné soustavě tak, že se mění jejich vzdálenost od osy otáčení. Coriolisova síla má směr kolmý ke spojnici těleso – osa otáčení a způsobuje stáčení trajektorie tělesa proti směru otáčení soustavy (v případě pohybu tělesa od středu otáčení. Pokud se těleso pohybuje směrem ke středu otáčení, působí síla ve směru otáčení. Například na těleso padající volným pádem nad rovníkem působí Coriolisova síla směrem k východu, tedy po směru otáčení). Patří mezi zdánlivé síly (nekoná práci). Je pojmenována po Gaspardu Gustavu de Coriolisovi, který se silami v rotujících soustavách zabýval v 19. století. Ovšem tento koncept již roku 1651 popsal i Giovanni Battista Riccioli.