  We consider finite area convex Euclidean circular sectors. We prove a
variational Polyakov formula which shows how the zeta-regularized determinant
of the Laplacian varies with respect to the opening angle. Varying the angle
corresponds to a conformal deformation in the direction of a conformal factor
with a logarithmic singularity at the origin. We compute explicitly all the
contributions to this formula coming from the different parts of the sector. In
the process, we obtain an explicit expression for the heat kernel on an
infinite area sector using Carslaw-Sommerfeld's heat kernel. We also compute
the zeta-regularized determinant of rectangular domains of unit area and prove
that it is uniquely maximized by the square.
