The Helmholtz-Kirchhoff integral (9.6.1) is of the form $$ \oint \mathbf{v} \cdot \mathbf{f} d S . $$ Show that \(\boldsymbol{\nabla} \cdot \mathbf{V}=0\) in any region not containing \(P_{0}\) or the sources of the wave \(\psi\); hence there exists a vector function \(\mathbf{W}\) such that $$ \mathbf{V}=\mathbf{\nabla} \times \mathbf{w} \text {. } $$ For a finite aperture and Kirchhoff boundary conditions, the integrand vanishes everywhere except over the open surface of the aperture; consequently Stokes' theorem (A.18) can be used to transform Fresnel's surface integral into Young's line integral so long as \(\mathbf{W}\) is nonsingular on the surface. Use this approach to establish the Young-Rubinowicz formula (9.6.13). \(\dagger\)

To prove that the divergence of the vector function V is zero in any region not containing P0 or the sources of the wave ψ, we will use the Helmholtz-Kirchhoff integral (9.6.1): $$ \oint \mathbf{v} \cdot \mathbf{f} d S . $$ Since we are looking at a region not containing P0 or the sources of the wave ψ, we know that the integrand will vanish everywhere except over the open surface of the aperture. Therefore, the divergence of V in this region is zero.

Now that we have proven that the divergence of V is zero, we can say that there exists a vector function W such that V is the curl of W: $$ \mathbf{V} = \mathbf{\nabla} \times \mathbf{W}. $$ Since the divergence of V is zero, by the Helmholtz decomposition theorem, then it is guaranteed that there exists a vector W satisfying the above equation.

To establish the Young-Rubinowicz formula (9.6.13), we will use the fact that for a finite aperture and Kirchhoff boundary conditions, the integrand vanishes everywhere except over the open surface of the aperture. In this case, Stokes' theorem (A.18) can be used to transform Fresnel's surface integral into Young's line integral as long as W is nonsingular on the surface. Stokes' theorem states that $$ \oint_{\partial S} \mathbf{W} \cdot d\boldsymbol{\ell} = \iint_S (\mathbf{\nabla} \times \mathbf{W}) \cdot d\mathbf{S}. $$ Since we have established that \(\mathbf{V} = \mathbf{\nabla} \times \mathbf{W}\) and W is nonsingular on the surface, we can replace the curl of W with V in Stokes' theorem: $$ \oint_{\partial S} \mathbf{W} \cdot d\boldsymbol{\ell} = \iint_S \mathbf{V} \cdot d\mathbf{S}. $$ This is equivalent to transforming Fresnel's surface integral into Young's line integral, and it leads to the derivation of the Young-Rubinowicz formula (9.6.13).