According to my former question "The gradient on sphere", the gradient of a function $f$ in spherical coordinates is given by \begin{align*} \nabla_{\mathbb{R}^3} f &= \frac{\partial f}{\partial r} \frac{\partial}{\partial r} + \nabla_{\mathbb{S}^2(r)} f, \end{align*} and the gradient on the $r$-sphere is $$ \nabla_{\mathbb{S}^2(r)} f = \frac{1}{r^2} \frac{\partial f}{\partial \varphi} \frac{\partial}{\partial \varphi} + \frac{1}{r^2 \sin \varphi} \frac{\partial f}{\partial \theta} \frac{\partial}{\partial \theta}. $$
However, according to the post "What is the metric tensor on the n-sphere (hypersphere)?", the metric of $\mathbb{S}^2(r)$ is given by $$ g_{\mathbb{S}^2(r)} = r^2 d\varphi \otimes d\varphi + r^2 \sin^2 \varphi \, d\theta \otimes d\theta. $$ Using this, we can calculate as follows: \begin{align*} \nabla_{\mathbb{S}^2(r)} f &= g^{11}_{\mathbb{S}^2(r)} \frac{\partial f}{\partial \varphi} \frac{\partial}{\partial \varphi} + g^{22}_{\mathbb{S}^2(r)} \frac{\partial f}{\partial \theta} \frac{\partial}{\partial \theta} \\ &= \frac{1}{r^2} \frac{\partial f}{\partial \varphi} \frac{\partial}{\partial \varphi} + \frac{1}{r^2 \sin^2 \varphi} \frac{\partial f}{\partial \theta} \frac{\partial}{\partial \theta}. \end{align*} I'm confused with the difference between the power of $\sin \varphi$. What went wrong?
Here is one more question. Can I write $\nabla_{\mathbb{S}^2(r)}f = \frac{1}{r} \nabla_{\mathbb{S}^2(1)} f$ if $f : \mathbb{R}^3 \setminus \lbrace 0 \rbrace \rightarrow \mathbb{R}$ satisfies $f(x) = f(x/ \vert x \vert)$?
