Kirchhoff's integral theorem (sometimes referred to as the Fresnel–Kirchhoff integral theorem) is a surface integral to obtain the value of the solution of the homogeneous scalar wave equation at an arbitrary point P in terms of the values of the solution and the solution's first-order derivative at all points on an arbitrary closed surface (on which the integration is performed) that encloses P. It is derived by using the Green's second identity and the homogeneous scalar wave equation that makes the volume integration in the Green's second identity zero. Integral Monochromatic wave The integral has the following form for a monochromatic wave: where the integration is performed over an arbitrary closed surface S enclosing the observation point , in is the wavenumber, in is the distance from an (infinitesimally small) integral surface element to the point , is the spatial part of the solution of the homogeneous scalar wave equation (i.e., as the homogeneous scalar wave equation solution), is the unit vector inward from and normal to the integral surface element, i.e., the inward surface normal unit vector, and denotes differentiation along the surface normal (i.e., a normal derivative) i.e., for a scalar field . Note that the surface normal is inward, i.e., it is toward the inside of the enclosed volume, in this integral; if the more usual outer-pointing normal is used, the integral will have the opposite sign. This integral can be written in a more familiar form where . Non-monochromatic wave A more general form can be derived for non-monochromatic waves. The complex amplitude of the wave can be represented by a Fourier integral of the form where, by Fourier inversion, we have The integral theorem (above) is applied to each Fourier component , and the following expression is obtained: where the square brackets on V terms denote retarded values, i.e. the values at time t − s/c. Kirchhoff showed that the above equation can be approximated to a simpler form in many cases, known as the Kirchhoff, or Fresnel–Kirchhoff diffraction formula, which is equivalent to the Huygens–Fresnel equation, except that it provides the inclination factor, which is not defined in the Huygens–Fresnel equation. The diffraction integral can be applied to a wide range of problems in optics. Integral derivation Here, the derivation of the Kirchhoff's integral theorem is introduced. First, the Green's second identity as the following is used. where the integral surface normal unit vector