Q 7 - Calculus

Question

. To convert from rectangular to cylindrical coordinates: 

$r=\_\_\_\_\_\_,\theta =\_\_\_\_\_,z=\_\_\_\_\_\_\_$

Step-by-Step Solution

Answer

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ \theta ={\tan }^{-1}\left(\frac{y}{x}\right) \\ z=z \end{gathered}$$

Step 1:Given information

Given (x,y,z) are rectangular cooedinates and $\left(r,\theta ,z\right)$ are cylindrical coordinates 

Step 2:Simplification

$\left(x,y,z\right)$are rectangular coordinates and $\left(r,\theta ,z\right)$ are cylindrical coordinates

The relation is as below: 

From the figure 

$\Rightarrow \cos \theta =\frac{x}{r},\sin \theta =\frac{y}{r}$

Now,

$\Rightarrow {\cos }^2\theta +{\sin }^2\theta =\frac{x^2}{r^2}+\frac{y^2}{r^2}$

$\Rightarrow r^2\left({\sin }^2\theta +{\cos }^2\theta \right)=x^2+y^2$

$\Rightarrow r^2=x^2+y^2$

$\Rightarrow r=\sqrt{x^2+y^2}$

Similarly

$\frac{\sin \theta }{\cos \theta }=\frac{y}{x}$

$\Rightarrow \tan \theta =\frac{y}{x}$

$\Rightarrow \theta ={\tan }^{-1}\frac{y}{x}$

And from the figure we can observe $z=z$