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Consider the surface $S$ in $\mathbb{R^3}$ given by the equation $$x^2 + y^2 + z^2 = 1$$

Let $φ ∈ [0, 2π)$ and $θ ∈ [0, π]$

Give a surface element $f : [0, 2π) × [0, π] → \mathbb{R^3}$ for $S$ in terms of the angles $φ, θ$

Could someone shove me in the right direction if possible? Need some confidence on the direction to head.

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    See [this](https://en.wikipedia.org/wiki/Spherical_coordinate_system#Cartesian_coordinates). – 2017-01-06
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    @ajotatxe, first time I realized that physicists swap the meanings of the angles in spherical coordinates. – 2017-01-06

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The surface $S$ is a sphere, since the equation $x^2+y^2+z^2=1$ describes a surface in $\Bbb R^3$ where all points have radial distance of $1$. One can use spherical coordinates to easily describe the surface $S$. The spherical to Cartesian coordinate conversion is as follows: $$\begin{cases}x=\rho\sin(\theta)\cos(\phi)\\y=\rho\sin(\theta)\sin(\phi)\\z=\rho\cos(\theta)\end{cases}$$ where $\theta\in[0,\pi]$, $\phi\in[0,2\pi)$, and $\rho^2=x^2+y^2+z^2$. So in the case of this sphere $S$, the radial distance is constant, so $\rho=1$. So the surface can be described by only the two parameters $\theta$ and $\phi$, which would have parametric equations as follows: $$\begin{cases}x(\phi,\theta)=\sin(\theta)\cos(\phi)\\y(\phi,\theta)=\sin(\theta)\sin(\phi)\\z(\phi,\theta)=\cos(\theta)\end{cases}$$

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    I wasnt sure if the question was asking for how we convert between coordinate systems or whether it wanted the integral. Does asking "Surface Element" not mean the integral with respect to $\rho,\theta,\phi$? Thanks for the clear answer! – 2017-01-06
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    @Clovers since in your question it says to find $f:[0,2\pi)\times[0,\pi]\to\Bbb R^3$ I assume that means to find a parameterization of the surface $S$ using just the parameters $\phi$ and $\theta$. Although that is ambiguous as I have not heard of such a parameterization being referred to as a "surface element". In my courses, surface element referred to $dS$ which is used in surface integral computations like $\int_S dS$. However, I think the question is asking to find a parameterization of the surface, because it does ask for that function $f$ in terms of $\phi$ and $\theta$. – 2017-01-06