$A = r \int_{0}^{\pi}\int_{0}^{2\pi} e^{i (\alpha+\theta)} d\alpha d\theta = r \int_{0}^{\pi} [\frac{-i}{\alpha+\theta} e^{i(\alpha + \theta)}]_{0}^{2\pi} d\theta = … = r \int_{0}^{\pi} e^{2i\alpha} ( \frac{1}{\alpha} – \frac{1}{\alpha+2\pi} ) d\theta$

and according to wolfram it is wrong, here. So how do you deduce the area of a sphere in polar coordinates?

[Edit] Trevor gave us formula $r^2 \int_{2π}^{0}\int^{π}_{0}sin(\phi) d\phi d\theta$ without deduction, by deduction, I mean things on which it is based on.

- Why is it odd function $sin(\phi)$ and not instead an even function $cos(\phi)$? (I know that it is and I can calculate it but some proof here missing)
- Why no term with $\theta$ but with $\phi$?
- Why $r^{2}$? -Okay it is area and by symmetry it sounds logical but why $r^{2}$ and not for example $rd$ where $d$ is some lenght or even a path?
- What about an arbitrary case?
- What determines the order in integration?

Many things open to really deduce it.

[Edit] Zhen Lin has an excellent answer where s/he noted that `2-sphere`

, i.e. the sphere in three dimensional euclidean space, is two-dimensional. Next deductions should be on `0-sphere`

and then incrementally on others. More about `n-sphere`

here.

- How do you deduce area in
`0-sphere`

,`1-sphere`

,`2-sphere`

, and`n-sphere`

? - Why does the Jacobian matrix in
`2-sphere-in-3-dim-euclidean-case`

have only two two variables $(\phi, \theta)$? - Are $\theta$ and $\phi$ non-free-variables because they are bounded somehow by the manifold (maybe misusing terminology)?
- What is $r$ in the sphere? For example in the
`2-sphere`

? - I can sense that the var $r$ is somehow different to $\phi$ and $\theta$ because what is it if you derivate it, zero so perhaps it is a trivial case (again probably misusing terminology, sorry)?
- What about if $r$ depends on $t$, time, what is it then? $\frac{\partial r}{\partial t}$ is now something. But would the extra dimension be now $t$ instead of $r$?

Please, use the term `dimension`

in your replies, I feel it is very important to even understand what a sphere is, let alone a dimension.

[Conclusion] The general solution is still open. The $r$ is a homothetic transformation. The terms $(\theta, \phi)$ are apparently tied to the manifold, (ALERT! perhaps misusing terminology). The acceptance of a question does not mean that all things appeared here are solved. Zhen`s solution is really the solution to look at, big thanks for it!