The integral is \begin{equation}\label{eq:20160227a} \int_S dx\ \frac{\vec n(x)}{|| y - x ||^{D-2} } = \begin{cases}\dfrac{D-2}{D} A y &\text{if }\quad ||y||\le 1 \\[5mm] \dfrac{D-2}{D} A \dfrac{y }{|| y ||^D}&\text{if }\quad ||y||\ge 1 \\ \end{cases} \end{equation} where

- \( D \ge 3 \)
- \( S \) is the sphere with radius \( 1 \) in \( \mathbb{R}^D \). The sphere itself has \( D-1 \) dimensions
- \( A \) is the area of this sphere
- \( \vec n \) is the unit vector perpendicular on the sphere pointing outwards

__Calculation__

Write \begin{equation*} F(y) = \int_S dx\ \frac{\vec n(x)}{|| y - x ||^{D-2} } \end{equation*} Because \begin{equation*} \Delta \dfrac{1}{|| x ||^{D-2} } = (2 - D ) A\ \delta(x) \end{equation*}

it follows that \begin{equation*} \Delta F(y) = \int_S dx\ \vec n(x) (2 - D ) A\ \delta(y-x) = \dfrac{y}{||y||} (2 - D ) A \int_S dx\ \delta(y-x) \end{equation*} Using \begin{equation*} \int_S dx\ \delta(y-x) = \delta( || y || - 1 ) \end{equation*} it follows that \(F\) satisfies the partial differential equation \begin{equation}\label{eq:20160227b} \Delta F(y) = y\ (2 - D )\ A\ \delta( || y || - 1 ) \end{equation} It is also clear that \( F(0) =0 \) and \( \lim_{y \to \infty} F(y) = 0 \). The integral \eqref{eq:20160227a} is thus given by the solution of the partial differential equation \eqref{eq:20160227b} subject to these boundary conditions. The easiest way to solve this partial differential equation is with the ansatz \begin{equation*} F(y) = y\ G(||y||) \end{equation*} The equation \eqref{eq:20160227b} then becomes \begin{equation*} y \Delta G +2 \frac{y}{||y||} G'(||y||) = y\ (2 - D )\ A\ \delta( || y || - 1 ) \end{equation*} Because \begin{equation*} \Delta G = \dfrac{1}{r^{D-1}} \partial_r \left( r^{D-1} \partial_r G \right) \end{equation*} the partial differential equation has been reduced to the ordinary differential equation \begin{equation*} \dfrac{1}{r^{D-1}} \partial_r \left( r^{D-1} \partial_r G \right) +2 \frac{1}{r} G'(r) = (2 - D )\ A\ \delta( r - 1 ) \end{equation*} which is the same as \begin{equation*} \dfrac{1}{r^{D+1}} \partial_r \left( r^{D-1} \partial_r G \right)= (2 - D )\ A\ \delta( r - 1 ) \end{equation*} This equation is easily solved. For example, the solution is given by the potential of a charged sphere in \( D + 2 \) dimensions. \begin{equation*} G(r) = \begin{cases}\dfrac{D-2}{D} A &\text{if }\quad r\le 1 \\[5mm] \dfrac{D-2}{D} A \dfrac{1 }{r^D}&\text{if }\quad r\ge 1 \\ \end{cases} \end{equation*} I find the calculation of the integral very long. Please leave a comment if you can calculate the integral in a quicker way or with a method that gives more insight.

## No comments:

## Post a Comment