Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Justify that $g$ is of class $\mathcal{C}^2$ on $\mathbb{R}^{*+} \times \mathbb{R}$.

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ For all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, express $\frac{\partial g}{\partial r}(r,\theta)$ and $\frac{\partial g}{\partial \theta}(r,\theta)$ in terms of $$\frac{\partial f}{\partial x}(r\cos(\theta), r\sin(\theta)) \quad \text{and} \quad \frac{\partial f}{\partial y}(r\cos(\theta), r\sin(\theta))$$

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Also express $\frac{\partial^2 g}{\partial r^2}(r,\theta)$ and $\frac{\partial^2 g}{\partial \theta^2}(r,\theta)$ in terms of the first and second partial derivatives of $f$ at $(r\cos(\theta), r\sin(\theta))$.

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Justify that $g$ is of class $\mathcal{C}^2$ on $\mathbb{R}^{*+} \times \mathbb{R}$.

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ For all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, express $\frac{\partial g}{\partial r}(r,\theta)$ and $\frac{\partial g}{\partial \theta}(r,\theta)$ in terms of $$\frac{\partial f}{\partial x}(r\cos(\theta), r\sin(\theta)) \quad \text{and} \quad \frac{\partial f}{\partial y}(r\cos(\theta), r\sin(\theta))$$

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Also express $\frac{\partial^2 g}{\partial r^2}(r,\theta)$ and $\frac{\partial^2 g}{\partial \theta^2}(r,\theta)$ in terms of the first and second partial derivatives of $f$ at $(r\cos(\theta), r\sin(\theta))$.