Polar Coordinates


PolarCoordinates
The polar coordinates r (the radial coordinate) and theta (the angular coordinate, often called the polar angle) are defined in terms of Cartesian coordinates by
x=rcostheta
(1)
y=rsintheta,
(2)
where r is the radial distance from the origin, and theta is the counterclockwise angle from the x-axis. In terms of x and y,
r=sqrt(x^2+y^2)
(3)
theta=tan^(-1)(y/x).
(4)
(Here, tan^(-1)(y/x) should be interpreted as the two-argument inverse tangent which takes the signs of x and y into account to determine in which quadrant theta lies.) It follows immediately that polar coordinates aren't inherently unique; in particular, (r,theta+2npi) will be precisely the same polar point as (r,theta) for any integer n. What's more, one often allows negative values of r under the assumption that (-r,theta) is plotted identically to (r,theta+/-pi).
PolarCoordinatesGraphPaper
The expression of a point as an ordered pair (r,theta) is known as polar notation, the equation of a curve expressed in polar coordinates is known as a polar equation, and a plot of a curve in polar coordinates is known as a polar plot.
In much the same way that Cartesian curves can be plotted on rectilinear axes, polar plots can be drawn on radialaxes such as those shown in the figure above.

The arc length of a polar curve given by r=r(theta) is
 s=int_(theta_1)^(theta_2)sqrt(r^2+((dr)/(dtheta))^2)dtheta.
(5)
The line element is given by
 ds^2=dr^2+r^2dtheta^2,
(6)
and the area element by
 dA=rdrdtheta.
(7)
The area enclosed by a polar curve r=r(theta) is
 A=1/2int_(theta_1)^(theta_2)r^2dtheta.
(8)
The slope of a polar function r=r(theta) at the point (r,theta) is given by
 m=(r+tantheta(dr)/(dtheta))/(-rtantheta+(dr)/(dtheta)).
(9)
The angle between the tangent and radial line at the point (r,theta) is
 psi=tan^(-1)(r/((dr)/(dtheta))).
(10)
A polar curve is symmetric about the x-axis if replacing theta by -theta in its equation produces an equivalent equation, symmetric about the y-axis if replacing theta by pi-theta in its equation produces an equivalent equation, and symmetric about the origin if replacing r by -r in its equation produces an equivalent equation.
In Cartesian coordinates, the radius vector r is
 r=sqrt(x^2+y^2)r^^,
(11)
giving derivative
 r^.=r^^^.sqrt(x^2+y^2)+r^^(x^2+y^2)^(-1/2)(xx^.+yy^.).
(12)
Its unit vector is
 r^^=(xx^^+yy^^)/(sqrt(x^2+y^2)),
(13)
giving derivative
 r^^^.=((xy^.-yx^.)(xy^^-yx^^))/((x^2+y^2)^(3/2)).
(14)
In polar coordinates, the radius vector is given by
 r=[rcostheta; rsintheta],
(15)
giving derivatives
r^.=[-rsinthetatheta^.+costhetar^.; rcosthetatheta^.+sinthetar^.]
(16)
=rtheta^.theta^^+r^.r^^
(17)
r^..=(r^..-rtheta^.^2)r^^+(2r^.theta^.+rtheta^..)theta^^
(18)
=(r^..-rtheta^.^2)r^^+1/rd/(dt)(r^2theta^.)theta^^.
(19)
The unit vectors are
r^^=((dr)/(dr))/(|(dr)/(dr)|)=[costheta; sintheta]
(20)
theta^^=((dr)/(dtheta))/(|(dr)/(dtheta)|)=[-sintheta; costheta],
(21)
giving derivatives
r^^^.=[-sinthetatheta^.; costhetatheta^.]=theta^.theta^^
(22)
theta^^^.=[-costhetatheta^.; -sinthetatheta^.]=-theta^.r^^.
(23)
By way of the Euler formula, the graphical representation of a complex number z=x+iy in terms of its complex modulus |z| and its complex argument theta is closely related to polar coordinates. Indeed, the Argand diagram of such a z in C is easily seen to be analogous to the usual polar plot.