Laplace equation in polar form

Aim. Transform u_xx + u_yy = 0 using x = r cos θ, y = r sin θ.

Step 1: relations.

r² = x² + y², θ = tan⁻¹(y/x)

∂r/∂x = x/r = cos θ        ∂r/∂y = y/r = sin θ
∂θ/∂x = −y/r² = −sin θ/r   ∂θ/∂y = x/r² = cos θ/r

Step 2: first derivatives (chain rule), u = u(r, θ).

u_x = u_r cos θ − u_θ sin θ / r
u_y = u_r sin θ + u_θ cos θ / r

Step 3: second derivatives. Apply the same operator ∂/∂x = cos θ ∂/∂r − (sin θ / r) ∂/∂θ to u_x:

u_xx = cos θ ∂/∂r (u_r cos θ − u_θ sin θ / r)
       − (sin θ / r) ∂/∂θ (u_r cos θ − u_θ sin θ / r)

     = cos²θ u_rr − 2 sinθ cosθ ( u_rθ / r − u_θ / r² )
       + (sin²θ / r²) u_θθ + (sin²θ / r) u_r

Similarly, with ∂/∂y = sin θ ∂/∂r + (cos θ / r) ∂/∂θ:

u_yy = sin²θ u_rr + 2 sinθ cosθ ( u_rθ / r − u_θ / r² )
       + (cos²θ / r²) u_θθ + (cos²θ / r) u_r

Step 4: add. The cross terms cancel and sin²θ + cos²θ = 1:

u_xx + u_yy = u_rr + (1/r) u_r + (1/r²) u_θθ

Answer: Laplace's equation in polar form is

∂²u/∂r² + (1/r) ∂u/∂r + (1/r²) ∂²u/∂θ² = 0

or equivalently r²u_rr + r u_r + u_θθ = 0.