Sistema de coordenadas polares

Two-dimensional coordinate system where each point is determined by a distance from reference point and an angle from a reference direction

Puntos en el sistema de coordenadas polares con el mástil de O y polar eje L . En verde, el punto con coordenada radial 3 y coordenada angular 60 grados o (3,  60 °). En azul, el punto (4,  210 °).

En matemáticas , el sistema de coordenadas polares es un sistema de coordenadas bidimensional en el que cada punto de un plano está determinado por una distancia desde un punto de referencia y un ángulo desde una dirección de referencia. El punto de referencia (análogo al origen de un sistema de coordenadas cartesiano ) se llama polo , y el rayo desde el polo en la dirección de referencia es el eje polar . La distancia desde el polo se llama coordenada radial , distancia radial o simplemente radio., y el ángulo se llama coordenada angular , ángulo polar o acimut . ^[1] La coordenada radial a menudo se denota por r o ρ , y la coordenada angular por φ , θ o t . Los ángulos en notación polar generalmente se expresan en grados o radianes (2 π rad es igual a 360 °).

Grégoire de Saint-Vincent y Bonaventura Cavalieri introdujeron de forma independiente los conceptos a mediados del siglo XVII, aunque el término actual coordenadas polares se ha atribuido a Gregorio Fontana en el siglo XVIII. La motivación inicial para la introducción del sistema polar fue el estudio del movimiento circular y orbital .

Las coordenadas polares son las más apropiadas en cualquier contexto donde el fenómeno que se está considerando está intrínsecamente ligado a la dirección y longitud desde un punto central en un plano, como las espirales . Los sistemas físicos planos con cuerpos que se mueven alrededor de un punto central, o fenómenos que se originan en un punto central, son a menudo más simples e intuitivos de modelar usando coordenadas polares.

El sistema de coordenadas polares se extiende a tres dimensiones de dos formas: los sistemas de coordenadas cilíndrico y esférico .

Historia

Ver también: Historia de las funciones trigonométricas

Hiparco

Los conceptos de ángulo y radio ya fueron utilizados por los pueblos antiguos del primer milenio antes de Cristo . El astrónomo y astrólogo griego Hiparco (190–120 a. C.) creó una tabla de funciones de las cuerdas dando la longitud de la cuerda para cada ángulo, y hay referencias a su uso de coordenadas polares para establecer posiciones estelares. ^[2] En Sobre espirales , Arquímedes describe la espiral de Arquímedes , una función cuyo radio depende del ángulo. Sin embargo, el trabajo griego no se extendió a un sistema de coordenadas completo.

Desde el siglo VIII d. C. en adelante, los astrónomos desarrollaron métodos para aproximar y calcular la dirección a La Meca ( qibla ) —y su distancia— desde cualquier lugar de la Tierra. ^[3] Desde el siglo IX en adelante, estaban usando trigonometría esférica y métodos de proyección de mapas para determinar estas cantidades con precisión. El cálculo es esencialmente la conversión de las coordenadas polares ecuatoriales de La Meca (es decir, su longitud y latitud ) a sus coordenadas polares (es decir, su qibla y distancia) en relación con un sistema cuyo meridiano de referencia es el gran círculo.a través de la ubicación dada y los polos de la Tierra y cuyo eje polar es la línea que pasa por la ubicación y su punto antípoda . ^[4]

Hay varios relatos de la introducción de coordenadas polares como parte de un sistema de coordenadas formal. La historia completa del tema se describe en El origen de las coordenadas polares del profesor de Harvard Julian Lowell Coolidge . ^[5] Grégoire de Saint-Vincent y Bonaventura Cavalieri introdujeron independientemente los conceptos a mediados del siglo XVII. Saint-Vincent escribió sobre ellos en privado en 1625 y publicó su trabajo en 1647, mientras que Cavalieri publicó el suyo en 1635 con una versión corregida que apareció en 1653. Cavalieri utilizó por primera vez coordenadas polares para resolver un problema relacionado con el área dentro de una espiral de Arquímedes . Blaise Pascalposteriormente utilizó coordenadas polares para calcular la longitud de los arcos parabólicos .

En Method of Fluxions (escrito en 1671, publicado en 1736), Sir Isaac Newton examinó las transformaciones entre las coordenadas polares, a las que se refirió como la "Séptima manera; para espirales", y otros nueve sistemas de coordenadas. ^[6] En la revista Acta Eruditorum (1691), Jacob Bernoulli utilizó un sistema con un punto en una línea, llamado polo y eje polar, respectivamente. Las coordenadas se especificaron por la distancia desde el polo y el ángulo desde el eje polar . El trabajo de Bernoulli se extendió hasta encontrar el radio de curvatura de las curvas expresadas en estas coordenadas.

El término actual coordenadas polares se ha atribuido a Gregorio Fontana y fue utilizado por escritores italianos del siglo XVIII. El término apareció en Inglés en George Peacock 's traducción de 1816 Lacroix ' s Cálculo Diferencial e Integral . ^{[7] [8]} Alexis Clairaut fue el primero en pensar en coordenadas polares en tres dimensiones, y Leonhard Euler fue el primero en desarrollarlas. ^[5]

Convenciones

Una cuadrícula polar con varios ángulos, aumentando en la orientación en sentido antihorario y etiquetada en grados

La coordenada radial a menudo se denota por r o ρ , y la coordenada angular por φ , θ o t . La coordenada angular se especifica como φ en la norma ISO 31-11 . Sin embargo, en la literatura matemática, el ángulo a menudo se denota por θ en lugar de φ .

Los ángulos en notación polar generalmente se expresan en grados o radianes (2 π rad es igual a 360 °). Los títulos se utilizan tradicionalmente en navegación , topografía y muchas disciplinas aplicadas, mientras que los radianes son más comunes en matemáticas y física matemática . ^[9]

El ángulo φ está definido para comenzar en 0 ° desde una dirección de referencia y aumentar para las rotaciones en la orientación en sentido antihorario (ccw) o en sentido horario (cw) . Por ejemplo, en matemáticas, la dirección de referencia generalmente se dibuja como un rayo desde el polo horizontalmente hacia la derecha, y el ángulo polar aumenta a ángulos positivos para las rotaciones en sentido antihorario, mientras que en la navegación ( rumbo , rumbo ) se dibuja el rumbo 0 °. verticalmente hacia arriba y el ángulo aumenta para las rotaciones de derecha a izquierda. Los ángulos polares disminuyen hacia valores negativos para rotaciones en las orientaciones respectivamente opuestas.

Unicidad de las coordenadas polares

Agregar cualquier número de vueltas completas (360 °) a la coordenada angular no cambia la dirección correspondiente. De manera similar, cualquier coordenada polar es idéntica a la coordenada con el componente radial negativo y la dirección opuesta (sumando 180 ° al ángulo polar). Por lo tanto, el mismo punto ( r , φ ) se puede expresar con un número infinito de coordenadas polares diferentes ( r , φ + n × 360 °) y (- r , φ + 180 ° + n × 360 °) = (- r , φ + (2 n + 1) × 180 °) , donde nes un entero arbitrario . ^[10] Además, el polo en sí se puede expresar como (0,  φ ) para cualquier ángulo φ . ^[11]

Cuando se necesita una representación única para cualquier punto además del polo, es habitual limitar r a números positivos ( r > 0 ) y φ al intervalo [0, 360 °) o al intervalo (−180 °, 180 °] , que en radianes son [0, 2π) o (−π, π] . ^[12] Otra convención, en referencia al codominio habitual de la función arctan , es permitir valores reales distintos de cero arbitrarios del componente radial y restringir el ángulo polar a (-90 °,  90 °] . En todos los casos, se debe elegir un acimut único para el polo ( r = 0), por ejemplo, φ = 0.

Conversión entre coordenadas polares y cartesianas

Un diagrama que ilustra la relación entre las coordenadas polares y cartesianas.

Una curva en el plano cartesiano se puede mapear en coordenadas polares. En esta animación, se asigna a . Haz clic en la imagen para los detalles.${\displaystyle y=\sin(6x)+2}$${\displaystyle r=\sin(6\theta )+2}$

Las coordenadas polares r y φ se pueden convertir a las coordenadas cartesianas x e y mediante el uso de la funciones trigonométricas seno y el coseno:

${\displaystyle {\begin{aligned}x&=r\cos \phi ,\\y&=r\sin \phi .\end{aligned}}}$

Las coordenadas cartesianas x e y pueden ser convertidos en coordenadas polares r y φ con r  ≥ 0 y φ en el intervalo (- π , π ] por: ^[13]

${\displaystyle r={\sqrt {x^{2}+y^{2}}}\quad }$(como en el teorema de Pitágoras o la norma euclidiana ), y

${\displaystyle \phi =\operatorname {atan2} (y,x),}$

donde atan2 es una variación común de la función arcotangente definida como

${\displaystyle \operatorname {atan2} (y,x)={\begin{cases}\arctan \left({\frac {y}{x}}\right)&{\mbox{if }}x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &{\mbox{if }}x<0{\mbox{ and }}y\geq 0\\\arctan \left({\frac {y}{x}}\right)-\pi &{\mbox{if }}x<0{\mbox{ and }}y<0\\{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y>0\\-{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y<0\\{\text{undefined}}&{\mbox{if }}x=0{\mbox{ and }}y=0.\end{cases}}}$

Si r se calcula primero como se indicó anteriormente, entonces esta fórmula para φ puede expresarse de manera un poco más simple usando la función de arcocoseno estándar :

${\displaystyle \phi ={\begin{cases}\arccos \left({\frac {x}{r}}\right)&{\mbox{if }}y\geq 0{\mbox{ and }}r\neq 0\\-\arccos \left({\frac {x}{r}}\right)&{\mbox{if }}y<0\\{\text{undefined}}&{\mbox{if }}r=0.\end{cases}}}$

El valor de φ anterior es el valor principal de la función de número complejo arg aplicada ax + iy . Se puede obtener un ángulo en el rango [0, 2 π ) sumando 2 π al valor en caso de que sea negativo (en otras palabras, cuando y es negativo).

Ecuación polar de una curva

La ecuación que define una curva algebraica expresada en coordenadas polares se conoce como ecuación polar . En muchos casos, dicha ecuación se puede especificar simplemente definiendo r como una función de φ . La curva resultante entonces consta de puntos de la forma ( r ( φ ),  φ ) y se puede considerar como la gráfica de la función polar r . Tenga en cuenta que, a diferencia de las coordenadas cartesianas, la variable independiente φ es la segunda entrada en el par ordenado.

Se pueden deducir diferentes formas de simetría de la ecuación de una función polar r . Si r (- φ ) = r ( φ ) la curva será simétrica con respecto al rayo horizontal (0 ° / 180 °), si r ( π - φ ) = r ( φ ) será simétrica con respecto a la vertical (90 °) / 270 °) rayo, y si r ( φ - α) = r ( φ ) será rotacionalmente simétrico por α en sentido horario y antihorario sobre el poste.

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

A circle with equation r(φ) = 1

The general equation for a circle with a center at (r₀, ${\displaystyle \gamma }$) and radius a is

${\displaystyle r^{2}-2rr_{0}\cos(\varphi -\gamma )+r_{0}^{2}=a^{2}.}$

This can be simplified in various ways, to conform to more specific cases, such as the equation

${\displaystyle r(\varphi )=a}$

for a circle with a center at the pole and radius a.^[14]

When r₀ = a, or when the origin lies on the circle, the equation becomes

${\displaystyle r=2a\cos(\varphi -\gamma ).}$

In the general case, the equation can be solved for r, giving

${\displaystyle r=r_{0}\cos(\varphi -\gamma )+{\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\varphi -\gamma )}},}$

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

${\displaystyle \varphi =\gamma ,}$

where ${\displaystyle \gamma }$ is the angle of elevation of the line; that is, ${\displaystyle \varphi =\arctan m}$, where ${\displaystyle m}$ is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line ${\displaystyle \varphi =\gamma }$ perpendicularly at the point ${\displaystyle (r_{0},\gamma )}$ has the equation

${\displaystyle r(\varphi )=r_{0}\sec(\varphi -\gamma ).}$

Otherwise stated ${\displaystyle (r_{0},\gamma )}$ is the point in which the tangent intersects the imaginary circle of radius ${\displaystyle r_{0}}$

Polar rose

A polar rose with equation r(φ) = 2 sin 4φ

A polar rose is a mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,

${\displaystyle r(\varphi )=a\cos \left(k\varphi +\gamma _{0}\right)}$

for any constant γ₀ (including 0). If k is an integer, these equations will produce a k-petaled rose if k is odd, or a 2k-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 γ₀ can be regarded as a phase angle.

Archimedean spiral

One arm of an Archimedean spiral with equation r(φ) = φ / 2π for 0 < φ < 6π

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

${\displaystyle r(\varphi )=a+b\varphi .}$

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 φ > 0 and one for φ < 0. The two arms are smoothly connected at the pole. If a = 0, 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

Ellipse, showing semi-latus rectum

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:

${\displaystyle r={\ell \over {1-e\cos \varphi }}}$

where e is the eccentricity and ${\displaystyle \ell }$ is the semi-latus rectum (the perpendicular distance at a focus from the major axis to the curve). If e > 1, this equation defines a hyperbola; if e = 1, it defines a parabola; and if e < 1, it defines an ellipse. The special case e = 0 of the latter results in a circle of the radius ${\displaystyle \ell }$.

Intersection of two polar curves

The graphs of two polar functions ${\displaystyle r=f(\theta )}$ and ${\displaystyle r=g(\theta )}$ have possible intersections of three types:

1. In the origin if the equations ${\displaystyle f(\theta )=0}$ and ${\displaystyle g(\theta )=0}$ have at least one solution each.
2. All the points ${\displaystyle [g(\theta _{i}),\theta _{i}]}$ where ${\displaystyle \theta _{i}}$ are the solutions to the equation ${\displaystyle f(\theta +2k\pi )=g(\theta )}$ where ${\displaystyle k}$ is an integer.
3. All the points ${\displaystyle [g(\theta _{i}),\theta _{i}]}$ where ${\displaystyle \theta _{i}}$ are the solutions to the equation ${\displaystyle f(\theta +(2k+1)\pi )=-g(\theta )}$ where ${\displaystyle k}$ is an integer.

Complex numbers

An illustration of a complex number z plotted on the complex plane

An illustration of a complex number plotted on the complex plane using Euler's formula

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

${\displaystyle z=x+iy}$

where i is the imaginary unit, or can alternatively be written in polar form (via the conversion formulae given above) as

${\displaystyle z=r(\cos \varphi +i\sin \varphi )}$

and from there as

${\displaystyle z=re^{i\varphi }}$

where e is Euler's number, which are equivalent as shown by Euler's formula.^[15] (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:

Multiplication

${\displaystyle r_{0}e^{i\varphi _{0}}\,r_{1}e^{i\varphi _{1}}=r_{0}r_{1}e^{i\left(\varphi _{0}+\varphi _{1}\right)}}$

Division

${\displaystyle {\frac {r_{0}e^{i\varphi _{0}}}{r_{1}e^{i\varphi _{1}}}}={\frac {r_{0}}{r_{1}}}e^{i(\varphi _{0}-\varphi _{1})}}$

Exponentiation (De Moivre's formula)

${\displaystyle \left(re^{i\varphi }\right)^{n}=r^{n}e^{in\varphi }}$

Calculus

Calculus can be applied to equations expressed in polar coordinates.^[16][17]

The angular coordinate φ is expressed in radians throughout this section, which is the conventional choice when doing calculus.

Differential calculus

Using x = r cos φ and y = r sin φ , 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)

${\displaystyle {\begin{aligned}r{\frac {\partial u}{\partial r}}&=r{\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial r}}+r{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial r}},\\[2pt]{\frac {\partial u}{\partial \varphi }}&={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial \varphi }}+{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial \varphi }},\end{aligned}}}$

or

${\displaystyle {\begin{aligned}r{\frac {\partial u}{\partial r}}&=r{\frac {\partial u}{\partial x}}\cos \varphi +r{\frac {\partial u}{\partial y}}\sin \varphi =x{\frac {\partial u}{\partial x}}+y{\frac {\partial u}{\partial y}},\\[2pt]{\frac {\partial u}{\partial \varphi }}&=-{\frac {\partial u}{\partial x}}r\sin \varphi +{\frac {\partial u}{\partial y}}r\cos \varphi =-y{\frac {\partial u}{\partial x}}+x{\frac {\partial u}{\partial y}}.\end{aligned}}}$

Hence, we have the following formulae:

${\displaystyle {\begin{aligned}r{\frac {\partial }{\partial r}}&=x{\frac {\partial }{\partial x}}+y{\frac {\partial }{\partial y}}\\[2pt]{\frac {\partial }{\partial \varphi }}&=-y{\frac {\partial }{\partial x}}+x{\frac {\partial }{\partial y}}.\end{aligned}}}$

Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function u(r,φ), it follows that

${\displaystyle {\begin{aligned}{\frac {\partial u}{\partial x}}&={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial x}}+{\frac {\partial u}{\partial \varphi }}{\frac {\partial \varphi }{\partial x}},\\[2pt]{\frac {\partial u}{\partial y}}&={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial y}}+{\frac {\partial u}{\partial \varphi }}{\frac {\partial \varphi }{\partial y}},\end{aligned}}}$

or

${\displaystyle {\begin{aligned}{\frac {\partial u}{\partial x}}&={\frac {\partial u}{\partial r}}{\frac {x}{\sqrt {x^{2}+y^{2}}}}-{\frac {\partial u}{\partial \varphi }}{\frac {y}{x^{2}+y^{2}}}\\[2pt]&=\cos \varphi {\frac {\partial u}{\partial r}}-{\frac {1}{r}}\sin \varphi {\frac {\partial u}{\partial \varphi }},\\[2pt]{\frac {\partial u}{\partial y}}&={\frac {\partial u}{\partial r}}{\frac {y}{\sqrt {x^{2}+y^{2}}}}+{\frac {\partial u}{\partial \varphi }}{\frac {x}{x^{2}+y^{2}}}\\[2pt]&=\sin \varphi {\frac {\partial u}{\partial r}}+{\frac {1}{r}}\cos \varphi {\frac {\partial u}{\partial \varphi }}.\end{aligned}}}$

Hence, we have the following formulae:

${\displaystyle {\begin{aligned}{\frac {\partial }{\partial x}}&=\cos \varphi {\frac {\partial }{\partial r}}-{\frac {1}{r}}\sin \varphi {\frac {\partial }{\partial \varphi }}\\[2pt]{\frac {\partial }{\partial y}}&=\sin \varphi {\frac {\partial }{\partial r}}+{\frac {1}{r}}\cos \varphi {\frac {\partial }{\partial \varphi }}.\end{aligned}}}$

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.

${\displaystyle {\begin{aligned}x&=r(\varphi )\cos \varphi \\y&=r(\varphi )\sin \varphi \end{aligned}}}$

Differentiating both equations with respect to φ yields

${\displaystyle {\begin{aligned}{\frac {dx}{d\varphi }}&=r'(\varphi )\cos \varphi -r(\varphi )\sin \varphi \\[2pt]{\frac {dy}{d\varphi }}&=r'(\varphi )\sin \varphi +r(\varphi )\cos \varphi .\end{aligned}}}$

Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point (r(φ), φ):

${\displaystyle {\frac {dy}{dx}}={\frac {r'(\varphi )\sin \varphi +r(\varphi )\cos \varphi }{r'(\varphi )\cos \varphi -r(\varphi )\sin \varphi }}.}$

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 0 < b − a < 2π. The length of L is given by the following integral

${\displaystyle L=\int _{a}^{b}{\sqrt {\left[r(\varphi )\right]^{2}+\left[{\tfrac {dr(\varphi )}{d\varphi }}\right]^{2}}}d\varphi }$

Integral calculus (area)

The integration region R is bounded by the curve r(φ) and the rays φ = a and φ = b.

Let R denote the region enclosed by a curve r(φ) and the rays φ = a and φ = b, where 0 < b − a ≤ 2π. Then, the area of R is

${\displaystyle {\frac {1}{2}}\int _{a}^{b}\left[r(\varphi )\right]^{2}\,d\varphi .}$

The region R is approximated by n sectors (here, n = 5).

A planimeter, which mechanically computes polar integrals

This result can be found as follows. First, the interval [a, b] is divided into n subintervals, where n is an arbitrary positive integer. Thus Δφ, the angle measure of each subinterval, is equal to b − a (the total angle measure of the interval), divided by n, the number of subintervals. For each subinterval i = 1, 2, ..., n, let φ_i be the midpoint of the subinterval, and construct a sector with the center at the pole, radius r(φ_i), central angle Δφ and arc length r(φ_i)Δφ. The area of each constructed sector is therefore equal to

${\displaystyle \left[r(\varphi _{i})\right]^{2}\pi \cdot {\frac {\Delta \varphi }{2\pi }}={\frac {1}{2}}\left[r(\varphi _{i})\right]^{2}\Delta \varphi .}$

Hence, the total area of all of the sectors is

${\displaystyle \sum _{i=1}^{n}{\tfrac {1}{2}}r(\varphi _{i})^{2}\,\Delta \varphi .}$

As the number of subintervals n is increased, the approximation of the area continues to improve. In the limit as n → ∞, 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:

${\displaystyle J=\det {\frac {\partial (x,y)}{\partial (r,\varphi )}}={\begin{vmatrix}{\frac {\partial x}{\partial r}}&{\frac {\partial x}{\partial \varphi }}\\[2pt]{\frac {\partial y}{\partial r}}&{\frac {\partial y}{\partial \varphi }}\end{vmatrix}}={\begin{vmatrix}\cos \varphi &-r\sin \varphi \\\sin \varphi &r\cos \varphi \end{vmatrix}}=r\cos ^{2}\varphi +r\sin ^{2}\varphi =r.}$

Hence, an area element in polar coordinates can be written as

${\displaystyle dA=dx\,dy\ =J\,dr\,d\varphi =r\,dr\,d\varphi .}$

Now, a function, that is given in polar coordinates, can be integrated as follows:

${\displaystyle \iint _{R}f(x,y)\,dA=\int _{a}^{b}\int _{0}^{r(\varphi )}f(r,\varphi )\,r\,dr\,d\varphi .}$

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.

The Gaussian integral

A more surprising application of this result yields the Gaussian integral, here denoted K:

${\displaystyle K=\int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}.}$

Vector calculus

Vector calculus can also be applied to polar coordinates. For a planar motion, let ${\displaystyle \mathbf {r} }$ be the position vector (r cos(φ), r sin(φ)), with r and φ depending on time t.

We define the unit vectors

${\displaystyle {\hat {\mathbf {r} }}=(\cos(\varphi ),\sin(\varphi ))}$

in the direction of ${\displaystyle \mathbf {r} }$ and

${\displaystyle {\hat {\boldsymbol {\varphi }}}=(-\sin(\varphi ),\cos(\varphi ))={\hat {\mathbf {k} }}\times {\hat {\mathbf {r} }}\ ,}$

in the plane of the motion perpendicular to the radial direction, where ${\displaystyle {\hat {\mathbf {k} }}}$ is a unit vector normal to the plane of the motion.

Then

${\displaystyle {\begin{aligned}\mathbf {r} &=(x,\ y)=r(\cos \varphi ,\ \sin \varphi )=r{\hat {\mathbf {r} }}\ ,\\{\dot {\mathbf {r} }}&=\left({\dot {x}},\ {\dot {y}}\right)={\dot {r}}(\cos \varphi ,\ \sin \varphi )+r{\dot {\varphi }}(-\sin \varphi ,\ \cos \varphi )={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\varphi }}{\hat {\boldsymbol {\varphi }}}\ ,\\{\ddot {\mathbf {r} }}&=\left({\ddot {x}},\ {\ddot {y}}\right)\\&={\ddot {r}}(\cos \varphi ,\ \sin \varphi )+2{\dot {r}}{\dot {\varphi }}(-\sin \varphi ,\ \cos \varphi )+r{\ddot {\varphi }}(-\sin \varphi ,\ \cos \varphi )-r{\dot {\varphi }}^{2}(\cos \varphi ,\ \sin \varphi )\\&=\left({\ddot {r}}-r{\dot {\varphi }}^{2}\right){\hat {\mathbf {r} }}+\left(r{\ddot {\varphi }}+2{\dot {r}}{\dot {\varphi }}\right){\hat {\boldsymbol {\varphi }}}\\&=\left({\ddot {r}}-r{\dot {\varphi }}^{2}\right){\hat {\mathbf {r} }}+{\frac {1}{r}}\;{\frac {d}{dt}}\left(r^{2}{\dot {\varphi }}\right){\hat {\boldsymbol {\varphi }}}.\end{aligned}}}$

Centrifugal and Coriolis terms

See also: Mechanics of planar particle motion and Centrifugal force (rotating reference frame)

Velocity vector v, always tangent to the path of motion.

Acceleration vector a, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.

Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2d space, but a plane in any higher dimension.

The term ${\displaystyle r{\dot {\varphi }}^{2}}$ is sometimes referred to as the centripetal acceleration, and the term ${\displaystyle 2{\dot {r}}{\dot {\varphi }}}$ as the Coriolis acceleration. For example, see Shankar.^[18]

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.

Inertial frame of reference S and instantaneous non-inertial co-rotating frame of reference S′. The co-rotating frame rotates at angular rate Ω equal to the rate of rotation of the particle about the origin of S′ at the particular moment t. Particle is located at vector position r(t) and unit vectors are shown in the radial direction to the particle from the origin, and also in the direction of increasing angle ϕ normal to the radial direction. These unit vectors need not be related to the tangent and normal to the path. Also, the radial distance r need not be related to the radius of curvature of the path.

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.^[19] 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 mrΩ², 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 −2m(dr/dt)Ω, pointed in the direction of increasing φ only. Thus, using these forces in Newton's second law we find:

${\displaystyle {\boldsymbol {F}}+{\boldsymbol {F}}_{\text{cf}}+{\boldsymbol {F}}_{\text{Cor}}=m{\ddot {\boldsymbol {r}}}\ ,}$

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:

${\displaystyle {\begin{aligned}F_{r}+mr\Omega ^{2}&=m{\ddot {r}}\\F_{\varphi }-2m{\dot {r}}\Omega &=mr{\ddot {\varphi }}\ ,\end{aligned}}}$

which can be compared to the equations for the inertial frame:

${\displaystyle {\begin{aligned}F_{r}&=m{\ddot {r}}-mr{\dot {\varphi }}^{2}\\F_{\varphi }&=mr{\ddot {\varphi }}+2m{\dot {r}}{\dot {\varphi }}\ .\end{aligned}}}$

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 ℝ² \ {(0,0)}, the plane minus the origin. In these coordinates, the Euclidean metric tensor is given by

$${\displaystyle ds^{2}=dr^{2}+r^{2}d\theta ^{2}.}$$

This 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 ds² = dx² + dy². An orthonormal frame with respect to this metric is given by

$${\displaystyle e_{r}={\frac {\partial }{\partial r}},\quad e_{\theta }={\frac {1}{r}}{\frac {\partial }{\partial \theta }},}$$

with dual coframe

$${\displaystyle e^{r}=dr,\quad e^{\theta }=rd\theta .}$$

The connection form relative to this frame and the Levi-Civita connection is given by the skew-symmetric matrix of 1-forms

$${\displaystyle \omega _{j}^{i}={\begin{pmatrix}0&-d\theta \\d\theta &0\end{pmatrix}}}$$

and 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.^[20] 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).^[21]

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 r = 0.5 + 0.5sin(ϕ) at its target design frequency.^[22] The pattern shifts toward omnidirectionality at lower frequencies.

See also

• Curvilinear coordinates
• List of canonical coordinate transformations
• Log-polar coordinates
• Polar decomposition
• Unit circle

References

1. Brown, Richard G. (1997). Andrew M. Gleason (ed.). Advanced Mathematics: Precalculus with Discrete Mathematics and Data Analysis. Evanston, Illinois: McDougal Littell. ISBN 0-395-77114-5.
2. Friendly, Michael (August 24, 2009). "Milestones in the History of Thematic Cartography, Statistical Graphics, and Data Visualization" (PDF).
3. King, David A. (2005). "The Sacred Geography of Islam". In Koetsier, Teun; Luc, Bergmans (eds.). Mathematics and the Divine: A Historical Study. Amsterdam: Elsevier. pp. 162–78. ISBN 0-444-50328-5.
4. King (2005, p. 169). The calculations were as accurate as could be achieved under the limitations imposed by their assumption that the Earth was a perfect sphere.
5. Coolidge, Julian (1952). "The Origin of Polar Coordinates". American Mathematical Monthly. Mathematical Association of America. 59 (2): 78–85. doi:10.2307/2307104. JSTOR 2307104.
6. Boyer, C. B. (1949). "Newton as an Originator of Polar Coordinates". American Mathematical Monthly. Mathematical Association of America. 56 (2): 73–78. doi:10.2307/2306162. JSTOR 2306162.
7. Miller, Jeff. "Earliest Known Uses of Some of the Words of Mathematics". Retrieved 2006-09-10.
8. Smith, David Eugene (1925). History of Mathematics, Vol II. Boston: Ginn and Co. p. 324.
9. Serway, Raymond A.; Jewett Jr., John W. (2005). Principles of Physics. Brooks/Cole—Thomson Learning. ISBN 0-534-49143-X.
10. "Polar Coordinates and Graphing" (PDF). 2006-04-13. Retrieved 2006-09-22.^{[permanent dead link]}
11. Lee, Theodore; David Cohen; David Sklar (2005). Precalculus: With Unit-Circle Trigonometry (Fourth ed.). Thomson Brooks/Cole. ISBN 0-534-40230-5.
12. Stewart, Ian; David Tall (1983). Complex Analysis (the Hitchhiker's Guide to the Plane). Cambridge University Press. ISBN 0-521-28763-4.
13. Torrence, Bruce Follett; Eve Torrence (1999). The Student's Introduction to Mathematica. Cambridge University Press. ISBN 0-521-59461-8.
14. Claeys, Johan. "Polar coordinates". Archived from the original on 2006-04-27. Retrieved 2006-05-25.
15. Smith, Julius O. (2003). "Euler's Identity". Mathematics of the Discrete Fourier Transform (DFT). W3K Publishing. ISBN 0-9745607-0-7. Archived from the original on 2006-09-15. Retrieved 2006-09-22.
16. Husch, Lawrence S. "Areas Bounded by Polar Curves". Retrieved 2006-11-25.
17. Lawrence S. Husch. "Tangent Lines to Polar Graphs". Retrieved 2006-11-25.
18. Ramamurti Shankar (1994). Principles of Quantum Mechanics (2nd ed.). Springer. p. 81. ISBN 0-306-44790-8.
19. For the following discussion, see John R Taylor (2005). Classical Mechanics. University Science Books. p. §9.10, pp. 358–359. ISBN 1-891389-22-X.
20. Santhi, Sumrit. "Aircraft Navigation System". Retrieved 2006-11-26.
21. "Emergency Procedures" (PDF). Archived from the original (PDF) on 2013-06-03. Retrieved 2007-01-15.
22. Eargle, John (2005). Handbook of Recording Engineering (Fourth ed.). Springer. ISBN 0-387-28470-2.

General references

• Adams, Robert; Christopher Essex (2013). Calculus: a complete course (Eighth ed.). Pearson Canada Inc. ISBN 978-0-321-78107-9.
• Anton, Howard; Irl Bivens; Stephen Davis (2002). Calculus (Seventh ed.). Anton Textbooks, Inc. ISBN 0-471-38157-8.
• Finney, Ross; George Thomas; Franklin Demana; Bert Waits (June 1994). Calculus: Graphical, Numerical, Algebraic (Single Variable Version ed.). Addison-Wesley Publishing Co. ISBN 0-201-55478-X.

External links

The Wikibook Calculus has a page on the topic of: Polar Integration

• "Polar coordinates", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Graphing Software at Curlie
• Coordinate Converter — converts between polar, Cartesian and spherical coordinates
• Polar Coordinate System Dynamic Demo