Question

Convert the vector A = xy ax - 2x ay to cylindrical system.

Answer 1

Given vector,

$A=xy(a_x-2xa_y)$

$\Rightarrow A= xya_x-2x^2ya_y$

Now, $x=a\cos\theta$

$y=a\sin\theta$

$z=z$

$B=a\cos\theta\times a\sin\theta\times a_y-2\times (a\cos\theta)^2\times a\sin\theta\times a_y$

we know that $a=a_x= a_y$ for the circle

$=a^3(cos\theta.\sin\theta-2\cos^2\theta.\sin\theta )$

$=a^3(\frac{1}{2}\sin2\theta-cos\theta. \sin2\theta)$

$=a^3\sin2\theta(\frac{1-2\cos\theta}{2})$

$=\frac{4a^3\sin2\theta \cos^2(\frac{\theta}{2})}{2}$

$=2a^3\sin2\theta \cos^2(\frac{\theta}{2})$