In the second edition of the book Introduction to Smooth Manifolds by John M. Lee, page 5,
EXAMPLE 1.4 (Spheres) : For each integer $n\geqslant 0$, the unite n-sphere $\mathbb{S}^{n}$ is Hausdorff and second-countable because it is a topological subspace of $\mathbb{R}^{n+1}$. To show that it is locally Euclidean, for each index $i = 1, \ldots n+1 $ let $U_{i}^{+}$ denote the subset of $\mathbb{R}^{n+1}$ where the ith coordinate is positive:
$U_{i}^{+} = \lbrace{(x_{1}, \ldots , x_{n})\in \mathbb{R}^{n+1} : x^{i} > 0 }\rbrace$
Let $f : \mathbb{B}^{n} \rightarrow \mathbb{R}$ be the continuos functions $f(u) = \sqrt{1 - \vert{u}\vert}$
Why $\,\,$ $U_{i}^{+}\cap\mathbb{S}^{n}$ is the graph of the function
$x^{i} = f(x^{1},\ldots,\hat{x^{i}}, \ldots,x^{n+1})$ ?
where the hat indicates that $x^{i}$ is omitted.