  We derive expressions, in terms of "polar shapelets", for the image
distortion operations associated with weak gravitational lensing. Shear causes
galaxy shapes to become elongated, and is sensitive to the second derivative of
the projected gravitational potential along their line of sight; flexion bends
galaxy shapes into arcs, and is sensitive to the third derivative. Polar
shapelets provide a natural representation, in which both shear and flexion
transformations are compact. Through this tool, we understand progress in
several weak lensing methods. We then exploit various symmetries of shapelets
to construct a range of shear estimators with useful properties. Through an
analogous investigation, we also explore several flexion estimators. In
particular, some of the estimators can be measured simultaneously and
independently for every galaxy, and will provide unique checks for systematics
in future weak lensing analyses. Using simulated images from the Shear TEsting
Programme (STEP), we show that we can recover input shears with no significant
bias. A complete software package to parametrize astronomical images in terms
of polar shapelets, and to perform a full weak lensing analysis, is available
at http://www.astro.caltech.edu/~rjm/shapelets/ .
