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Let $f(r,\theta)=(r\cos\theta ,r\sin\theta)$ for $(r,\theta)$ $\in \mathbb R^2$ with $r\ne0$. then which are true statements?

$1$.$Df(r,\theta)$ is not zero for any $(r,\theta)$ with $r\ne0$

$2$.$Df(r,\theta)=r^2I$ for any $(r,\theta)$ with $r\ne0$

here $Df(r,\theta)$ is the matrix $$\begin{bmatrix} \cos\theta &-r \sin\theta \\ \sin\theta & r\cos\theta \end{bmatrix}$$.its determinant is $r$ which is nonzero.so $1$ true.but what about $2$?

Can anyone help me please .

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For 1, consider the following: Can $$\cos \theta=\sin \theta=0?$$ For 2, consider the following example: Let $$(r,\theta)=(r,\pi).$$

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