$\displaystyle f(z) = u + iv

u = u(x,y)

v = v(x,y)$

use chain rule and

$\displaystyle x = rcos (\theta)

y = r sin (\theta)$

to show

$\displaystyle u_x = u_r * cos (\theta) - u_\theta * sin(\theta)/r$

$\displaystyle v_x = v_r * cos(\theta) - v_\theta * sin(\theta)/r$

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what i know is, if i had to find $\displaystyle u_r$ then i would go about:

$\displaystyle u_r = u_x * u_r + u_y * y_r$

plugging values gives

$\displaystyle u_r = u_x * cos(\theta) + u_y * sin(\theta)$

similarly can find [/tex] u_\theta , v_r and v_theta[/tex]

but to find $\displaystyle u_x$, how would i set up the chain rule to begin with? In above case, x and y are function of r and theta so it makes sense. ho.wever, here i can't do similar with u_x...i think.

please show full work if possible.