# Thread: chain rule in complex numbers

1. ## chain rule in complex numbers

$f(z) = u + iv

u = u(x,y)
v = v(x,y)$

use chain rule and
$x = rcos (\theta)
y = r sin (\theta)$

to show

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

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

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

$u_r = u_x * u_r + u_y * y_r$

plugging values gives

$u_r = u_x * cos(\theta) + u_y * sin(\theta)$
similarly can find [/tex] u_\theta , v_r and v_theta[/tex]

but to find $u_x$, 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.

2. For this problem, consider that you could also define $r,\theta$ as functions of $x,y$, like so:

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

$\theta = \tan^{-1}\left( \frac{y}{x} \right)$

Then the chain rule looks like this:

$u_x = u_r r_x + u_\theta \theta_x$

Think you can finish it from here?