Question

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

Answer 1

`(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)" ")|)`

Answer 2

`(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`.

Answer 3

`(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.