Determine the expression of the Riemannnian metric of $S^2$ induced by $\mathbb{R}^3$ 
Consider usual local coordinates  $(\theta, \varphi)$ em $S^2 \subset \mathbb{R}^3$ defined by the parametrization $\phi:(0, \pi) \times (0, 2 \pi) \rightarrow \mathbb{R}^3$ given by
$$\phi(\theta, \varphi) = (\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta)$$
Using those coordinates determine the expression of the riemannian metric induced over $S^2$ by the euclidian metric of $\mathbb{R}^3$.

Conceptually the riemannian metric will associate each point $p \in S^2$ to a product $\langle \cdot,\cdot \rangle_p$ in its tangent space $T_pM$. I think that what the question means by metric induced by $\mathbb{R}^3$ is taking $\langle \cdot,\cdot \rangle_p$ to be the usual internal product in $\mathbb{R}^3$. I'm having trouble with getting an explicit expression for the metric.

Essentially, the metric will have the form $${\rm d}s^2 = E(\theta,\varphi){\rm d}\theta^2 +2F(\theta,\varphi){\rm d}\theta\,{\rm d}\varphi + G(\theta,\varphi){\rm d}\varphi^2,$$where $$E(\theta,\phi)=\left\langle\frac{\partial\phi}{\partial \theta}(\theta,\varphi),\frac{\partial\phi}{\partial \theta}(\theta,\varphi)\right\rangle,\quad F(\theta,\varphi)=\left\langle\frac{\partial\phi}{\partial \theta}(\theta,\varphi),\frac{\partial\phi}{\partial \varphi}(\theta,\varphi)\right\rangle,\quad\mbox{and}\quad G(\theta,\varphi)=\left\langle\frac{\partial\phi}{\partial \varphi}(\theta,\varphi),\frac{\partial\phi}{\partial \varphi}(\theta,\varphi)\right\rangle,$$where these inner products are computed with the ambient metric. This is a general mechanism: if $(M^n,g)$ is a Riemannian manifold, $\iota:S \to M$ is a submanifold, and $(u^1,...,u^k)$ are coordinates for $S$, then $\iota^*g$ is described in these coordinates as $$\sum_{i,j=1}^k a_{ij}\,{\rm d}u^i\,{\rm d}u^j,$$where $$a_{ij} = g\left({\rm d}\iota\left(\frac{\partial}{\partial u^i}\right),{\rm d}\iota\left(\frac{\partial}{\partial u^j}\right)\right).$$
