  We have developed a new technique called Direct Shear Mapping (DSM) to
measure gravitational lensing shear directly from observations of a single
background source. The technique assumes the velocity map of an un-lensed,
stably-rotating galaxy will be rotationally symmetric. Lensing distorts the
velocity map making it asymmetric. The degree of lensing can be inferred by
determining the transformation required to restore axisymmetry. This technique
is in contrast to traditional weak lensing methods, which require averaging an
ensemble of background galaxy ellipticity measurements, to obtain a single
shear measurement. We have tested the efficacy of our fitting algorithm with a
suite of systematic tests on simulated data. We demonstrate that we are in
principle able to measure shears as small as 0.01. In practice, we have fitted
for the shear in very low redshift (and hence un-lensed) velocity maps, and
have obtained null result with an error of $\pm 0.01$. This high sensitivity
results from analysing spatially resolved spectroscopic images (i.e. 3D data
cubes), including not just shape information (as in traditional weak lensing
measurements) but velocity information as well. Spirals and rotating
ellipticals are ideal targets for this new technique. Data from any large IFU
or radio telescope is suitable, or indeed any instrument with spatially
resolved spectroscopy such as SAMI, ALMA, HETDEX and SKA.
