= Solution
One convention for the <Papkovich–Neuber representation> uses a harmonic vector potential $\boldsymbol\Phi$ and a harmonic scalar potential $\chi$:
$$
\boxed{2\mu\mathbf u=\nabla(\mathbf r\cdot\boldsymbol\Phi+\chi)-2\boldsymbol\Phi,\qquad p=\nabla\cdot\boldsymbol\Phi,\qquad\nabla^2\boldsymbol\Phi=0,\quad\nabla^2\chi=0.}
$$
These are <harmonic functions> away from any singular <force> point. Since $\nabla^2(\mathbf r\cdot\boldsymbol\Phi)=2\nabla\cdot\boldsymbol\Phi$, the representation gives $\nabla\cdot\mathbf u=0$ and $\mu\nabla^2\mathbf u=\nabla p$, the equations of homogeneous <Stokes flow>.
Place the point <force> at the origin. A <velocity> linear in $\mathbf F$, decaying as $r^{-1}$ and having the rotational symmetry of a point <force> is obtained from $\boldsymbol\Phi=c\mathbf F/r$, $\chi=0$. The vector components are <harmonic functions> for $r>0$; scalar dipole potentials would instead generate higher-order decaying singularities. The <force> normalization fixes $c=-1/(4\pi)$. Indeed, substitution gives the <Stokeslet>:
$$
\boxed{\mathbf u(\mathbf r)=\frac1{8\pi\mu}\left(\frac{\mathbf F}{r}+\frac{(\mathbf F\cdot\mathbf r)\mathbf r}{r^3}\right),\qquad p(\mathbf r)=\frac{\mathbf F\cdot\mathbf r}{4\pi r^3}.}
$$
To verify its strength, the <Newtonian fluid stress tensor> is $\sigma_{ij}=-3(\mathbf F\cdot\mathbf r)r_i r_j/(4\pi r^5)$. Its outward traction integrated over any <sphere> surrounding the origin is $-\mathbf F$, because $\int\mathbf n\mathbf n\,d\Omega=(4\pi/3)I$. Thus the localized <force> applied to the fluid is $\mathbf F$, as required. Translation of the origin gives the same <Stokeslet> centered at any prescribed <force> point.
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