Polar Coordinates
In mathematics, the polar coordinate system is a twodimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction.
The fixed point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the fixed direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is the angular coordinate, polar angle, or azimuth.^{[1]}
Contents
History
The concepts of angle and radius were already used by ancient peoples of the 1st millennium BCE. The Greek astronomer and astrologer Hipparchus (190–120 BCE) 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.^{[2]} 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 CE onward, astronomers developed methods for approximating and calculating the direction to Makkah (qibla)—and its distance—from any location on the Earth.^{[3]} 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.^{[4]}
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.^{[5]} Grégoire de SaintVincent and Bonaventura Cavalieri independently introduced the concepts in the midseventeenth century. SaintVincent 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.^{[6]} 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 18thcentury Italian writers. The term appeared in English in George Peacock's 1816 translation of Lacroix's Differential and Integral Calculus.^{[7]}^{[8]} Alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them.^{[5]}
Conventions
The radial coordinate is often denoted by r, and the angular coordinate by φ, θ or t. The angular coordinate is specified as φ by ISO standard 3111.
Angles in polar notation are generally expressed in either degrees or radians (2π 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.^{[9]}
In many contexts, a positive angular coordinate means that the angle φ is measured counterclockwise from the axis.
In mathematical literature, the polar axis is often drawn horizontal and pointing to the right.
Uniqueness of polar coordinates
Adding any number of full turns (360°) to the angular coordinate does not change the corresponding direction. Also, a negative radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the same point can be expressed with an infinite number of different polar coordinates (r, φ ± n×360°) or (−r, φ ± (2n + 1)180°), where n is any integer.^{[10]} Moreover, the pole itself can be expressed as (0, φ) for any angle φ.^{[11]}
Where a unique representation is needed for any point, it is usual to limit r to nonnegative numbers (r ≥ 0) and φ to the interval [0, 360°) or (−180°, 180°] (in radians, [0, 2π) or (−π, π]).^{[12]} One must also choose a unique azimuth for the pole, 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:
 $x\; =\; r\; \backslash cos\; \backslash varphi\; \backslash ,$
 $y\; =\; r\; \backslash sin\; \backslash varphi\; \backslash ,$
The Cartesian coordinates x and y can be converted to polar coordinates r and φ with r ≥ 0 and φ in the interval (−π, π] by:^{[13]}
 $r\; =\; \backslash sqrt\{x^2\; +\; y^2\}\; \backslash quad$ (as in the Pythagorean theorem or the Euclidean norm), and
 $\backslash varphi\; =\; \backslash operatorname\{atan2\}(y,\; x)\; \backslash quad$,
where atan2 is a common variation on the arctangent function defined as
 $\backslash operatorname\{atan2\}(y,\; x)\; =$
\begin{cases} \arctan(\frac{y}{x}) & \mbox{if } x > 0\\ \arctan(\frac{y}{x}) + \pi & \mbox{if } x < 0 \mbox{ and } y \ge 0\\ \arctan(\frac{y}{x})  \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}
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, 2π) may be obtained by adding 2π to the value in case it 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.
Different forms of symmetry can be deduced from the equation of a polar function r. If r(−φ) = r(φ) the curve will be symmetrical about the horizontal (0°/180°) ray, if r(π − φ) = r(φ) it will be symmetric about the vertical (90°/270°) ray, and if r(φ − α) = r(φ) it will be rotationally symmetric α 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 (r_{0}, $\backslash gamma$) and radius a is
 $r^2\; \; 2\; r\; r\_0\; \backslash cos(\backslash varphi\; \; \backslash gamma)\; +\; r\_0^2\; =\; a^2.\backslash ,$
This can be simplified in various ways, to conform to more specific cases, such as the equation
 $r(\backslash varphi)=a\; \backslash ,$
for a circle with a center at the pole and radius a.^{[14]}
When r_{0} = a, or when the origin lies on the circle, the equation becomes
 $r\; =\; 2\; a\backslash cos(\backslash varphi\; \; \backslash gamma)$.
In the general case, the equation can be solved for r, giving
 $r\; =\; r\_0\; \backslash cos(\backslash varphi\; \; \backslash gamma)\; +\; \backslash sqrt\{a^2\; \; r\_0^2\; \backslash sin^2(\backslash varphi\; \; \backslash gamma)\}$,
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
 $\backslash varphi\; =\; \backslash gamma\; \backslash ,$,
where ɣ is the angle of elevation of the line; that is, ɣ = arctan m where m is the slope of the line in the Cartesian coordinate system. The nonradial line that crosses the radial line φ = ɣ perpendicularly at the point (r_{0}, ɣ) has the equation
 $r(\backslash varphi)\; =\; \{r\_0\}\backslash sec(\backslash varphi\backslash gamma).\; \backslash ,$
Otherwise stated (r_{0}, ɣ) is the point in which the tangent intersects the imaginary circle of radius r_{0}.
Polar rose
A polar rose is a famous mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,
 $r(\backslash varphi)\; =\; a\; \backslash cos\; (k\backslash varphi\; +\; \backslash gamma\_0)\backslash ,$
for any constant ɣ_{0} (including 0). If k is an integer, these equations will produce a kpetaled rose if k is odd, or a 2kpetaled rose if k is even. If k is rational but not an integer, a roselike 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 represents the length of the petals of the rose.
Archimedean spiral
The Archimedean spiral is a famous spiral that was discovered by Archimedes, which also can be expressed as a simple polar equation. It is represented by the equation
 $r(\backslash varphi)\; =\; a+b\backslash varphi.\; \backslash ,$
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. 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:
 $r\; =\; \{\; \backslash ell\backslash over\; \{1\; +\; e\; \backslash cos\; \backslash varphi\}\}$
where e is the eccentricity and $\backslash ell$ is the semilatus 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 radius $\backslash ell$.
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
 $z\; =\; x\; +\; iy\backslash ,$
where i is the imaginary unit, or can alternatively be written in polar form (via the conversion formulae given above) as
 $z\; =\; r\backslash cdot(\backslash cos\backslash varphi+i\backslash sin\backslash varphi)$
and from there as
 $z\; =\; re^\{i\backslash varphi\}\; \backslash ,$
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:
 $r\_0\; e^\{i\backslash varphi\_0\}\; \backslash cdot\; r\_1\; e^\{i\backslash varphi\_1\}=r\_0\; r\_1\; e^\{i(\backslash varphi\_0\; +\; \backslash varphi\_1)\}\; \backslash ,$
 Division:
 $\backslash frac\{r\_0\; e^\{i\backslash varphi\_0\}\}\{r\_1\; e^\{i\backslash varphi\_1\}\}=\backslash frac\{r\_0\}\{r\_1\}e^\{i(\backslash varphi\_0\; \; \backslash varphi\_1)\}\; \backslash ,$
 Exponentiation (De Moivre's formula):
 $(re^\{i\backslash varphi\})^n=r^ne^\{in\backslash varphi\}\; \backslash ,$
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
 $r\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; r\}\; =\; r\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\}\backslash frac\{\backslash partial\; x\}\{\backslash partial\; r\}\; +\; r\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; y\}\backslash frac\{\backslash partial\; y\}\{\backslash partial\; r\},$
 $\backslash frac\{\backslash partial\; u\}\{\backslash partial\; \backslash varphi\}\; =\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\}\backslash frac\{\backslash partial\; x\}\{\backslash partial\; \backslash varphi\}\; +\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; y\}\backslash frac\{\backslash partial\; y\}\{\backslash partial\; \backslash varphi\},$
or
 $r\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; r\}\; =\; r\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\}\; \backslash cos\; \backslash varphi\; +\; r\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; y\}\; \backslash sin\; \backslash varphi\; =\; x\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\}\; +\; y\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; y\},$
 $\backslash frac\{\backslash partial\; u\}\{\backslash partial\; \backslash varphi\}\; =\; \; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\}\; r\; \backslash sin\; \backslash varphi\; +\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; y\}\; r\; \backslash cos\; \backslash varphi\; =\; y\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\}\; +\; x\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; y\}.$
Hence, we have the following formulae:
 $r\; \backslash frac\{\backslash partial\}\{\backslash partial\; r\}=\; x\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; +\; y\; \backslash frac\{\backslash partial\}\{\backslash partial\; y\}\; \backslash ,$
 $\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash varphi\}\; =\; y\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; +\; x\; \backslash frac\{\backslash partial\}\{\backslash partial\; y\}\; .$
Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function u(r,φ), it follows that
 $\backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\}\; =\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; r\}\backslash frac\{\backslash partial\; r\}\{\backslash partial\; x\}\; +\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; \backslash varphi\}\backslash frac\{\backslash partial\; \backslash varphi\}\{\backslash partial\; x\},$
 $\backslash frac\{\backslash partial\; u\}\{\backslash partial\; y\}\; =\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; r\}\backslash frac\{\backslash partial\; r\}\{\backslash partial\; y\}\; +\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; \backslash varphi\}\backslash frac\{\backslash partial\; \backslash varphi\}\{\backslash partial\; y\},$
or
 $\backslash frac\{\backslash partial\; u\}\{\backslash partial\; x\}\; =\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; r\}\backslash frac\{x\}\{\backslash sqrt\{x^2+y^2\}\}\; \; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; \backslash varphi\}\backslash frac\{y\}\{x^2+y^2\}\; =\; \backslash cos\; \backslash varphi\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; r\}\; \; \backslash frac\{1\}\{r\}\; \backslash sin\; \backslash varphi\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; \backslash varphi\},$
 $\backslash frac\{\backslash partial\; u\}\{\backslash partial\; y\}\; =\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; r\}\backslash frac\{y\}\{\backslash sqrt\{x^2+y^2\}\}\; +\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; \backslash varphi\}\backslash frac\{x\}\{x^2+y^2\}\; =\; \backslash sin\; \backslash varphi\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; r\}\; +\; \backslash frac\{1\}\{r\}\; \backslash cos\; \backslash varphi\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; \backslash varphi\}.$
Hence, we have the following formulae:
 $\backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; =\; \backslash cos\; \backslash varphi\; \backslash frac\{\backslash partial\}\{\backslash partial\; r\}\; \; \backslash frac\{1\}\{r\}\; \backslash sin\; \backslash varphi\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash varphi\}\; \backslash ,$
 $\backslash frac\{\backslash partial\}\{\backslash partial\; y\}\; =\; \backslash sin\; \backslash varphi\; \backslash frac\{\backslash partial\}\{\backslash partial\; r\}\; +\; \backslash frac\{1\}\{r\}\; \backslash cos\; \backslash varphi\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash varphi\}.$
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.
 $x=r(\backslash varphi)\backslash cos\backslash varphi\; \backslash ,$
 $y=r(\backslash varphi)\backslash sin\backslash varphi\; \backslash ,$
Differentiating both equations with respect to φ yields
 $\backslash frac\{dx\}\{d\backslash varphi\}=r\text{'}(\backslash varphi)\backslash cos\backslash varphir(\backslash varphi)\backslash sin\backslash varphi\; \backslash ,$
 $\backslash frac\{dy\}\{d\backslash varphi\}=r\text{'}(\backslash varphi)\backslash sin\backslash varphi+r(\backslash varphi)\backslash cos\backslash varphi.\; \backslash ,$
Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point (r(φ), φ):
 $\backslash frac\{dy\}\{dx\}=\backslash frac\{r\text{'}(\backslash varphi)\backslash sin\backslash varphi+r(\backslash varphi)\backslash cos\backslash varphi\}\{r\text{'}(\backslash varphi)\backslash cos\backslash varphir(\backslash varphi)\backslash sin\backslash varphi\}.$
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
 $L\; =\; \backslash int\_a^b\; \backslash sqrt\{\; \backslash left[r(\backslash varphi)\backslash right]^2\; +\; \backslash left[\; \backslash right]\; ^2\; \}\; d\backslash varphi$
Integral calculus (area)
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
 $\backslash frac12\backslash int\_a^b\; \backslash left[r(\backslash varphi)\backslash right]^2\backslash ,\; d\backslash varphi.$
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 length of each subinterval, is equal to b − a (the total length 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
 $\backslash left[r(\backslash varphi\_i)\backslash right]^2\; \backslash pi\; \backslash cdot\; \backslash frac\{\backslash Delta\; \backslash varphi\}\{2\backslash pi\}\; =\; \backslash frac\{1\}\{2\}\backslash left[r(\backslash varphi\_i)\backslash right]^2\; \backslash Delta\; \backslash varphi.$
Hence, the total area of all of the sectors is
 $\backslash sum\_\{i=1\}^n\; \backslash tfrac12r(\backslash varphi\_i)^2\backslash ,\backslash Delta\backslash varphi.$
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 2element 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:
 $J\; =\; \backslash det\backslash frac\{\backslash partial(x,y)\}\{\backslash partial(r,\backslash varphi)\}$
=\begin{vmatrix}
\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \varphi} \\[8pt] \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
 $dA\; =\; dx\backslash ,dy\backslash \; =\; J\backslash ,dr\backslash ,d\backslash varphi\; =\; r\backslash ,dr\backslash ,d\backslash varphi.$
Now, a function that is given in polar coordinates can be integrated as follows:
 $\backslash iint\_R\; f(x,y)\; \backslash ,\; dA\; =\; \backslash int\_a^b\; \backslash int\_0^\{r(\backslash varphi)\}\; f(r,\backslash varphi)\backslash ,r\backslash ,dr\backslash ,d\backslash varphi.$
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. A more surprising application of this result yields the Gaussian integral
 $\backslash int\_\{\backslash infty\}^\backslash infty\; e^\{x^2\}\; \backslash ,\; dx\; =\; \backslash sqrt\backslash pi.$
Vector calculus
Vector calculus can also be applied to polar coordinates. For a planar motion, let $\backslash mathbf\{r\}$ be the position vector (rcos(φ), rsin(φ)), with r and φ depending on time t.
We define the unit vectors
 $\backslash hat\{\backslash mathbf\{r\}\}=(\backslash cos(\backslash varphi),\backslash sin(\backslash varphi))$
in the direction of r and
 $\backslash hat\{\backslash boldsymbol\backslash varphi\}=(\backslash sin(\backslash varphi),\backslash cos(\backslash varphi))\; =\; \backslash hat\; \{\backslash mathbf\{k\}\}\; \backslash times\; \backslash hat\; \{\backslash mathbf\{r\}\}\; \backslash \; ,$
in the plane of the motion perpendicular to the radial direction, where $\backslash hat\{\backslash mathbf\; \{k\}\}$ is a unit vector normal to the plane of the motion.
Then
 $\backslash mathbf\{r\}\; =\; (x,\; \backslash \; y\; )\; =\; r\; (\backslash cos\; \backslash varphi\; ,\backslash \; \backslash sin\; \backslash varphi)\; =\; r\; \backslash hat\{\backslash mathbf\{r\}\}\backslash \; ,$
 $\backslash dot\; \{\backslash mathbf\; r\}\; =\; (\backslash dot\; x,\; \backslash \; \backslash dot\; y\; )\; =\; \backslash dot\; r\; (\backslash cos\; \backslash varphi\; ,\backslash \; \backslash sin\; \backslash varphi)\; +\; r\; \backslash dot\; \backslash varphi\; (\backslash sin\; \backslash varphi\; ,\backslash \; \backslash cos\; \backslash varphi)\; =\; \backslash dot\; r\; \backslash hat\; \{\backslash mathbf\; r\}\; +\; r\; \backslash dot\; \backslash varphi\; \backslash hat\; \{\backslash boldsymbol\{\backslash varphi\}\}\; \backslash \; ,$
 $\backslash ddot\; \{\backslash mathbf\; r\; \}\; =\; (\backslash ddot\; x,\; \backslash \; \backslash ddot\; y\; )\; =\; \backslash ddot\; r\; (\backslash cos\; \backslash varphi\; ,\backslash \; \backslash sin\; \backslash varphi)\; +\; 2\backslash dot\; r\; \backslash dot\; \backslash varphi\; (\backslash sin\; \backslash varphi\; ,\backslash \; \backslash cos\; \backslash varphi)\; +\; r\backslash ddot\; \backslash varphi\; (\backslash sin\; \backslash varphi\; ,\backslash \; \backslash cos\; \backslash varphi)\; \; r\; \{\backslash dot\; \backslash varphi\; \}^2\; (\backslash cos\; \backslash varphi\; ,\backslash \; \backslash sin\; \backslash varphi)\backslash \; =$
 $\backslash left(\; \backslash ddot\; r\; \; r\backslash dot\backslash varphi^2\; \backslash right)\; \backslash hat\{\backslash mathbf\; r\}\; +\; \backslash left(\; r\backslash ddot\backslash varphi\; +\; 2\backslash dot\; r\; \backslash dot\backslash varphi\; \backslash right)\; \backslash hat\backslash boldsymbol\{\backslash varphi\}\; \backslash \; =\; (\backslash ddot\; r\; \; r\backslash dot\backslash varphi^2)\backslash hat\{\backslash mathbf\{r\}\}\; +\; \backslash frac\{1\}\{r\}\backslash ;\; \backslash frac\{d\}\{dt\}\; \backslash left(r^2\backslash dot\backslash varphi\backslash right)\; \backslash hat\{\backslash boldsymbol\backslash varphi\}$
Centrifugal and Coriolis terms
The term $r\backslash dot\backslash varphi^2$ is sometimes referred to as the centrifugal term, and the term $2\backslash dot\; r\; \backslash dot\backslash varphi$ as the Coriolis term. For example, see Shankar.^{[18]} Although these equations bear some resemblance in form to the centrifugal and Coriolis effects found in rotating reference frames, nonetheless these are not the same things.^{[19]} For example, the physical centrifugal and Coriolis forces appear only in noninertial frames of reference. In contrast, these terms that appear when acceleration is expressed in polar coordinates are a mathematical consequence of differentiation; these terms appear wherever polar coordinates are used. In particular, these terms appear even when polar coordinates are used in inertial frames of reference, where the physical centrifugal and Coriolis forces never appear.
Corotating frame
For a particle in planar motion, one approach to attaching physical significance to these terms is based on the concept of an instantaneous corotating frame of reference.^{[20]} To define a corotating 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 corotating 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 corotating frame. Let the location of the particle in the inertial frame be (r(t), φ(t)), and in the corotating frame be (r(t), φ′(t)). Because the corotating frame rotates at the same rate as the particle, dφ′/dt = 0. The fictitious centrifugal force in the corotating frame is mrΩ^{2}, radially outward. The velocity of the particle in the corotating 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:
 $\backslash boldsymbol\{F\}\; +\; \backslash boldsymbol\{F\_\{cf\}\}\; +\; \backslash boldsymbol\{F\_\{Cor\}\}\; =\; m\; \backslash ddot\{\backslash boldsymbol\{r\}\}\; \backslash \; ,$
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:
 $F\_r\; +\; mr\backslash Omega^2\; =\; m\backslash ddot\; r$
 $F\_\{\backslash varphi\}2m\backslash dot\; r\; \backslash Omega\; =\; mr\; \backslash ddot\; \{\backslash varphi\}\; \backslash \; ,$
which can be compared to the equations for the inertial frame:
 $F\_r\; =\; m\; \backslash ddot\; r\; mr\; \backslash dot\; \{\backslash varphi\}^2\; \backslash $
 $F\_\{\backslash varphi\}\; =\; mr\; \backslash ddot\; \backslash varphi\; +2m\; \backslash dot\; r\; \backslash dot\; \{\backslash varphi\}\; \backslash \; .$
This comparison, plus the recognition that by the definition of the corotating 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, noninertial corotating 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.
Connection to spherical and cylindrical coordinates
The polar coordinate system is extended into three dimensions with two different coordinate systems, the cylindrical and spherical coordinate system.
Applications
Polar coordinates are twodimensional and thus they can be used only where point positions lie on a single twodimensional 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.
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.^{[21]} Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read zeroninerzero by air traffic control).^{[22]}
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 inversesquare 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.^{[23]} The pattern shifts toward omnidirectionality at lower frequencies.
See also
References
 General
 Specific
External links
The Wikibook Calculus has a page on the topic of: Polar Integration 
 Template:Springer
 DMOZ
 Coordinate Converter — converts between polar, Cartesian and spherical coordinates
 Polar Coordinate System Dynamic Demo
