What is the polar form of ( 0,0 )?

3 Answers
Aug 14, 2017

(0,0)

Explanation:

Well, this coordinate point is the origin of both the Cartesian and polar coordinate planes, so the polar form is also color(blue)((0,0).

Nevertheless, we can solve this if we'd like, using the equations

  • r^2 = x^2 + y^2

  • theta = arctan(y/x)

We know:

  • x=0

  • y=0

So we have

r^2 = 0^2 + 0^2

ul(r = 0

theta by convention will be 0, because in reality the arctangent calculation is undefined (it has no specific direction, so we say the angle is 0).

The coordinate is thus

color(blue)(ulbar(|stackrel(" ")(" "(0,0)" ")|)

Aug 14, 2017

(r,theta) -= (0,0).


Well, consider the conversion:

x = rcostheta
y = rsintheta

(x,y) = (0,0) = (rcostheta, rsintheta)

We necessarily have that

0 = rcostheta = rsintheta

If sintheta = 0, it is never true that costheta = 0 for the same theta. In fact, you can convince yourself that:

  • if sintheta = 0, costheta = pm1, and...
  • if costheta = 0, sintheta = pm1.

Thus, we must have that r = 0 for the equality to be satisfied.

However, even when r = 0, we note that theta could take on the usual angles (0^@, . . . , ), allowing an infinite number of polar coordinates that correspond to a given Cartesian origin, i.e.

overbrace((0","0))^((x","y)) harr overbrace((0","theta))^((r","theta))

Despite that, since the distance from the origin is zero, we can choose theta = 0^@, since we describe the same spatial coordinates with any theta when r = 0.

Aug 14, 2017

(0,0), most of the time !

Explanation:

By convention we choose (0,0) as the polar origin, or "pole". However as indicated in the other solutions the polar origin is not unique, as (0,pi), (0,pi/2) or in fact (0,phi) AA phi in RR all represent the polar origin.

Ordinarily the ambiguity does not represent an issue, however one must take extreme caution when converting from rectangular to polar coordinates in integration, as the Jacobian:

J = | ( (partial r)/(partial x), (partial r)/(partial y) ), ( (partial theta)/(partial x), (partial theta)/(partial y) ) |

is not defined at the origin.

Similarly if we are examining limits at the origin we need to be very careful about the value of theta to use and we should not assume that (0,0) is appropriate.