Calculus

Let $z = f(x, y)$ be a differentiable function where $x = r \cos \theta$ and $y = r \sin \theta$. Which of the following expressions are mathematically CORRECT?

Options:

• (A) $\left(\frac{\partial z}{\partial r}\right)^2 + \frac{1}{r^2}\left(\frac{\partial z}{\partial \theta}\right)^2 = \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2$
• (B) $\frac{\partial z}{\partial r} = \frac{\partial z}{\partial x} \cos \theta + \frac{\partial z}{\partial y} \sin \theta$
• (C) $\frac{\partial z}{\partial \theta} = -r \frac{\partial z}{\partial x} \sin \theta + r \frac{\partial z}{\partial y} \cos \theta$
• (D) $\frac{\partial^2 z}{\partial r^2} + \frac{1}{r} \frac{\partial z}{\partial r} + \frac{1}{r^2} \frac{\partial^2 z}{\partial \theta^2} = \frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2}$