Vectorial Analysis , convert from Cartesian (x,y,z) to Cylindrical (ρ,θ,z)

Question

r= $5t^2 $ $\sqrt{1-t^2}$ i +$5t^3$j + $3t^2$ k

I was trying to convert that exercice and find speed and acceleration in cilindrycal coordinates but somehow I got lost because what I have

$x=pcos\theta$; $y=psen\theta$; z=z

and $e\theta$ =-$sen\theta$i+$cos\theta$j, $ep$ =$cos\theta$i+$sen\theta$j ez=ez

When I try to convert I always got messy results with $cos\theta$ and $sen\theta$

REsult: r= $10tep$i + $5t^2 $ / $\sqrt{1-t^2}$$e$$\phi$j + 6$tk$

I would appreciate your help.

Answer 1

We have that

• $x=5t^2\sqrt{1-t^2}$
• $y=5t^3$
• $z=3t^2$

then

• $\rho=\sqrt{x^2+y^2}=\sqrt{25t^4(1-t^2)+25t^6}=5t^2$

since $x$ is always positive

• $\theta = \arctan \frac{y}{x}=\arctan \frac{t}{\sqrt{1-t^2}}$