Chain Rule — Question 5

Question 5 – Solution

Strategy. Treat $r$ and $\theta$ as independent inputs to the Cartesian coordinate map.

Step 1: General formulas $$\boxed{u_r=u_x\cos\theta+u_y\sin\theta},$$ $$\boxed{u_\theta=-r u_x\sin\theta+r u_y\cos\theta}.$$

Step 2: Apply For $u=x^2-y^2$, $u_x=2x$, $u_y=-2y$. Substituting $x=r\cos\theta$, $y=r\sin\theta$ gives $$u_r=2r(\cos^2\theta-\sin^2\theta)=\boxed{2r\cos 2\theta},$$ $$u_\theta=-4r^2\sin\theta\cos\theta=\boxed{-2r^2\sin 2\theta}.$$

Step 3: Verify Directly, $u=r^2\cos 2\theta$; its two ordinary partial derivatives match both formulas.