### 3.6 Lens mapping

In the vicinity of an arbitrary point, the lens mapping as shown in Equation (7) can be described by its
Jacobian matrix :
Here we made use of the fact (see [26, 164]), that the deflection angle can be expressed as the gradient of
an effective two-dimensional scalar potential : , where
and is the Newtonian potential of the lens.
The determinant of the Jacobian is the inverse of the magnification:

Let us define
The Laplacian of the effective potential is twice the convergence:
With the definitions of the components of the external shear ,
and
(where the angle reflects the direction of the shear-inducing tidal force relative to the coordinate
system), the Jacobian matrix can be written
The magnification can now be expressed as a function of the local convergence and the local shear :
Locations at which have formally infinite magnification. They are called critical curves in the
lens plane. The corresponding locations in the source plane are the caustics. For spherically symmetric
mass distributions, the critical curves are circles. For a point lens, the caustic degenerates into a point. For
elliptical lenses or spherically symmetric lenses plus external shear, the caustics can consist of cusps and
folds. In Figure 4 the caustics and critical curves for an elliptical lens with a finite core are
displayed.