<s>[INST] Is the intersection of $x^2 + y^2 + z^2 = 1$ and $x = \frac{1}{2}$ a manifold in $\mathbb{R}^3$? Is the intersection of $x^2 + y^2 + z^2 = 1$ and $x = \frac{1}{2}$ a manifold in $\mathbb{R}^3$?
I think that it is, because it can be parameterized by $f(x) = (\frac{1}{2},\sqrt{\frac{3}{4}} \cos x, \sqrt{\frac{3}{4}} \sin x)$ for $0 < x < 2\pi$.
Is this correct?
 [/INST] Let
$$f:\Bbb R^3\to\Bbb R^2,\quad f(x,y,z)=(x^2+y^2+z^2-1,x-1/2).$$
Then, your space is $f^{-1}(0)$. Now,
$$df=\begin{pmatrix}
2x & 2y & 2z \\
1 & 0 & 0
\end{pmatrix}
$$
which has full rank for all $(x,y,z)\in f^{-1}(0)$, so yes it is a manifold.
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