Editing N-sphere (section)

===Spherical volume and area elements===
The arc length element is<math display="block">d s^2=d r^2+\sum_{k=1}^{n-1} r^2\left(\prod_{m=1}^{k-1} \sin ^2\left(\varphi_m\right)\right) d \varphi_k^2</math>To express the [[volume element]] of {{tmath|n}}-dimensional Euclidean space in terms of spherical coordinates, let {{tmath|s_k {{=}} \sin \varphi_k}} and {{tmath|c_k {{=}} \cos \varphi_k}} for concision, then observe that the [[Jacobian matrix and determinant|Jacobian matrix]] of the transformation is:
:<math>
J_n =
\begin{pmatrix} 
c_1                        &-rs_1              &0        &0 &\cdots &0      \\
s_1c_2                     &rc_1c_2            &-rs_1s_2 &0 &\cdots &0      \\
\vdots                     &\vdots             & \vdots  &  &\ddots &\vdots \\
                           &                   &         &  &       &0      \\
s_1\cdots s_{n-2}c_{n-1}   &\cdots             &\cdots   &  &       &-rs_1\cdots s_{n-2}s_{n-1} \\
s_{1}\cdots s_{n-2}s_{n-1} &rc_1\cdots s_{n-1} &\cdots   &  &       &\phantom{-}rs_1\cdots s_{n-2}c_{n-1}
\end{pmatrix}.
</math>

The determinant of this matrix can be calculated by induction.  When {{tmath|n {{=}} 2}}, a straightforward computation shows that the determinant is {{tmath|r}}.  For larger {{tmath|n}}, observe that {{tmath|J_n}} can be constructed from {{tmath|J_{n-1} }} as follows.  Except in column {{tmath|n}}, rows {{tmath|n-1}} and {{tmath|n}} of {{tmath|J_n}} are the same as row {{tmath|n-1}} of {{tmath|J_{n-1} }}, but multiplied by an extra factor of {{tmath|\cos \varphi_{n-1} }} in row {{tmath|n-1}} and an extra factor of {{tmath|\sin \varphi_{n-1} }} in row {{tmath|n}}.  In column {{tmath|n}}, rows {{tmath|n-1}} and {{tmath|n}} of {{tmath|J_n}} are the same as column {{tmath|n-1}} of row {{tmath|n-1}} of {{tmath|J_{n-1} }}, but multiplied by extra factors of {{tmath|\sin \varphi_{n-1} }} in row {{tmath|n-1}} and {{tmath|\cos \varphi_{n-1} }} in row {{tmath|n}}, respectively.  The determinant of {{tmath|J_n}} can be calculated by [[Laplace expansion]] in the final column.  By the recursive description of {{tmath|J_n}}, the submatrix formed by deleting the entry at {{tmath|(n-1, n)}} and its row and column almost equals {{tmath|J_{n-1} }}, except that its last row is multiplied by {{tmath|\sin \varphi_{n-1} }}.  Similarly, the submatrix formed by deleting the entry at {{tmath|(n, n)}} and its row and column almost equals {{tmath|J_{n-1} }}, except that its last row is multiplied by {{tmath|\cos \varphi_{n-1} }}.  Therefore the determinant of {{tmath|J_n}} is
:<math>\begin{align}
|J_n|
&= (-1)^{(n-1)+n}(-rs_1 \dotsm s_{n-2}s_{n-1})(s_{n-1}|J_{n-1}|) \\
&\qquad {}+ (-1)^{n+n}(rs_1 \dotsm s_{n-2}c_{n-1})(c_{n-1}|J_{n-1}|) \\
&= (rs_1 \dotsm s_{n-2}|J_{n-1}|(s_{n-1}^2 + c_{n-1}^2) \\
&= (rs_1 \dotsm s_{n-2})|J_{n-1}|.
\end{align}</math>
Induction then gives a [[closed-form expression]] for the volume element in spherical coordinates
:<math>\begin{align}
d^nV &= 
\left|\det\frac{\partial (x_i)}{\partial\left(r,\varphi_j\right)}\right|
dr\,d\varphi_1 \, d\varphi_2\cdots d\varphi_{n-1} \\
&= r^{n-1}\sin^{n-2}(\varphi_1)\sin^{n-3}(\varphi_2)\cdots \sin(\varphi_{n-2})\,
dr\,d\varphi_1 \, d\varphi_2\cdots d\varphi_{n-1}.
\end{align}</math>
The formula for the volume of the {{tmath|n}}-ball can be derived from this by integration.

Similarly the surface area element of the {{tmath|(n-1)}}-sphere of radius {{tmath|r}}, which generalizes the [[area element]] of the {{tmath|2}}-sphere, is given by

: <math>d_{S^{n-1}}V = R^{n-1}\sin^{n-2}(\varphi_1)\sin^{n-3}(\varphi_2)\cdots \sin(\varphi_{n-2})\, d\varphi_1 \, d\varphi_2\cdots d\varphi_{n-1}.</math>

The natural choice of an orthogonal basis over the angular coordinates is a product of [[Gegenbauer polynomial|ultraspherical polynomials]],

: <math>\begin{align}
& {} \quad \int_0^\pi \sin^{n-j-1}\left(\varphi_j\right) C_s^{\left(\frac{n-j-1}{2}\right)}\cos \left(\varphi_j \right)C_{s'}^{\left(\frac{n-j-1}{2}\right)}\cos \left(\varphi_j\right) \, d\varphi_j \\[6pt]
& = \frac{2^{3-n+j}\pi \Gamma(s+n-j-1)}{s!(2s+n-j-1)\Gamma^2\left(\frac{n-j-1}{2}\right)}\delta_{s,s'}
\end{align}</math>

for {{tmath|j {{=}} 1, 2, \ldots, n-2}}, and the {{tmath|e^{is\varphi_j} }} for the angle {{tmath|j {{=}} n-1}} in concordance with the [[spherical harmonics]].