Consider an open set U in the Euclidean space Rⁿ and a continuous function u defined on U with real or complex values. Let x be a point in U and r > 0 be such that the closed ball B(x, r) of center x and radius r is contained in U. The spherical mean over the sphere of radius r centered at x is defined as

${\displaystyle {\frac {1}{\omega _{n-1}(r)}}\int \limits _{\partial B(x,r)}\!u(y)\,\mathrm {d} S(y)}$ 

where ∂B(x, r) is the (n − 1)-sphere forming the boundary of B(x, r), dS denotes integration with respect to spherical measure and ω_n−1(r) is the "surface area" of this (n − 1)-sphere.

Equivalently, the spherical mean is given by

${\displaystyle {\frac {1}{\omega _{n-1}}}\int \limits _{\|y\|=1}\!u(x+ry)\,\mathrm {d} S(y)}$ 

where ω_n−1 is the area of the (n − 1)-sphere of radius 1.

The spherical mean is often denoted as

${\displaystyle \int \limits _{\partial B(x,r)}\!\!\!\!\!\!\!\!\!-\,u(y)\,\mathrm {d} S(y).}$ 

The spherical mean is also defined for Riemannian manifolds in a natural manner.

• From the continuity of ${\displaystyle u}$  it follows that the function
  $${\displaystyle r\to \int \limits _{\partial B(x,r)}\!\!\!\!\!\!\!\!\!-\,u(y)\,\mathrm {d} S(y)}$$

   
  is continuous, and that its limit as ${\displaystyle r\to 0}$  is ${\displaystyle u(x).}$ 
• Spherical means can be used to solve the Cauchy problem for the wave equation ${\displaystyle \partial _{t}^{2}u=c^{2}\,\Delta u}$  in odd space dimension. The result, known as Kirchhoff's formula, is derived by using spherical means to reduce the wave equation in ${\displaystyle \mathbb {R} ^{n}}$  (for odd ${\displaystyle n}$ ) to the wave equation in ${\displaystyle \mathbb {R} }$ , and then using d'Alembert's formula. The expression itself is presented in wave equation article.
• If ${\displaystyle U}$  is an open set in ${\displaystyle \mathbb {R} ^{n}}$  and ${\displaystyle u}$  is a C² function defined on ${\displaystyle U}$ , then ${\displaystyle u}$  is harmonic if and only if for all ${\displaystyle x}$  in ${\displaystyle U}$  and all ${\displaystyle r>0}$  such that the closed ball ${\displaystyle B(x,r)}$  is contained in ${\displaystyle U}$  one has
  $${\displaystyle u(x)=\int \limits _{\partial B(x,r)}\!\!\!\!\!\!\!\!\!-\,u(y)\,\mathrm {d} S(y).}$$

   
  This result can be used to prove the maximum principle for harmonic functions.

• Evans, Lawrence C. (1998). Partial differential equations. American Mathematical Society. ISBN 978-0-8218-0772-9.

• Sabelfeld, K. K.; Shalimova, I. A. (1997). Spherical means for PDEs. VSP. ISBN 978-90-6764-211-8.
• Sunada, Toshikazu (1981). "Spherical means and geodesic chains in a Riemannian manifold". Trans. Am. Math. Soc. 267 (2): 483–501. doi:10.1090/S0002-9947-1981-0626485-6.

• Spherical mean at PlanetMath.