Q. 5 - Calculus

Question

This problem concerns the connections between the polar plane and the θr-plane. 

(a) Which points in the θr-plane correspond to the pole in the polar plane, and why? 

(b) Which points in the θr-plane correspond to the horizontal axis in the polar plane, and why?

 (c) Which points in the θr-plane correspond to the vertical axis in the polar plane, and why? 

Step-by-Step Solution

Answer

a)$\left(\theta ,0\right)$ corresponds to the pole.

b) $\left(0,r\right),\left(\mathrm{\pi },-\mathrm{r}\right)$ corresponds to the points on the horizontal axis.

c)$\left(\frac{\mathrm{\pi }}{2},r\right),\left(\frac{3\mathrm{\pi }}{2},-r\right)$ corresponds to points on the vertical axis.

Step 1. Given information

Two different planes:  polar plane and $\theta r$ plane.

Part (a) Step 1. Find the points in the θr-plane correspond to the pole in the polar plane

The pole in polar form is the point where $r=0$

So the point in $\theta r$ plane is $\left(0,0\right)$

Part (b) Step 1. Find points in the θr-plane correspond to the horizontal axis in the polar plane:

In the polar plain the points on horizontal axis is $\left(r,0\right),\left(-r,\mathrm{\pi }\right)$

In $\theta r$ plane these points correspond to point $\left(0,r\right),\left(\mathrm{\pi },-\mathrm{r}\right)$

Part (c) Step 1. Find points in the θr-plane correspond to the vertical axis in the polar plane.

In polar plane the points on vertical lines are $\left(r,\frac{\mathrm{\pi }}{2}\right),\left(-r,\frac{3\mathrm{\pi }}{2}\right)$

In $\theta r$ plane that points corresponds to the point $\left(\frac{\mathrm{\pi }}{2},r\right), \left(\frac{3\mathrm{\pi }}{2},-r\right)$