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The irradiance at a point on a surface due to a polyhedral source of uniform brightness is given by a well-known analytic formula. In this paper we derive the corresponding analytic expression for the irradiance Jacobian, the derivative of the vector representation of irradiance. Although the result is elementary for unoccluded sources, within penumbrae the irradiance Jacobian must incorporate more information about blockers than either the irradiance or vector irradiance. The expression presented here holds for any number of polyhedral blockers and requires only a minor extension of standard polygon clipping to evaluate. To illustrate its use, three related applications are briefing described: direct computation of isolux contours, finding local irradiance extrema, and iso-meshing. Isolux contours are curves of constant irradiance across a surface that can be followed using a predictor-corrector method based on the irradiance Jacobian. Similarly, local extrema can be found using a descent method. Finally, iso-meshing is a new approach to surface mesh generation that incorporates families of isolux contours.