This paper presents a theoretical (closed-form) solution for the z-axis surface deformations of a linear, homogeneous, unconstrained and isotropic paraboloidal surface subjected to a 3-dimensional linear thermal temperature gradient and soak temperature change. Previously, an equation for the component of the nodal surface displacement in the z direction has been published. Attaching the z-axis component of the nodal surface displacement to the original surface does not accurately describe the final surface. This work extends the previous analysis and presents a polynomial equation for the corrected surface deformation along the z-axis, as well as, the coefficients for the standard Zernike polynomial describing the corrected surface deformation. Also included is a discussion about z-axis temperature gradients across the paraboloidal surface and how to calculate an equivalent soak temperature change.