chain rule for partial derivatives

[hholzer]

If I have u = u(x,y) and let (r, t) be polar coordinates, then
expressing u_x and u_y in terms of u_r and u_t could be
done with a system of linear equations in u_x and u_y?

I get:
u_r = u_x * x_r + u_y * y_r

u_t = u_x * x_t + u_y * y_t

is this the right direction? Because by substitution,
I end up with:
(u_t* x_r - u_r)/(y_t*x_r - y_r) = u_y which
does not seem right, considering:
u_y = u_r * r_y + u_t * t_y

Any insight to the problem is appreciated.

[hamster143]

No, that should be

(u_t x_r - u_r x_t ) / (y_t x_r - x_t y_r) = u_y

and since

x_r = cos t
y_r = sin t

x_t = -r sin t
y_t = r cos t

it implies that u_y = (u_t cos t + u_r r sin t) / (r cos t cos t + r sin t sin t) = u_t cos t / r + u_r sin t

[hholzer]

I found my error. Thanks for your response!

One question:
How did you determine the following:
x_t = -r sin t
y_t = r cos t