Lensing near critical points

Abstract

The lens equation, derived in Chap. 4 and discussed in some detail in the previous chapter, defines a surjective mapping $f:{{\mathbb{R}}^{2}} \to {{\mathbb{R}}^{2}},x \mapsto y$ from the lens plane to the source plane. We assume here that it is differentiable as often as is needed. If ${{y}^{{(0)}}} = f({\text{ }}{{x}^{{(0)}}})$ is the image of x ⁽⁰⁾) and if the Jacobian D = det A of the derivative $A = \frac{{\partial y}}{{\partial x}} = \frac{{\partial ({\text{ }}{{y}_{1}},{\text{ }}{{y}_{2}})}}{{\partial ({\text{ }}{{x}_{1}},{\text{ }}{{x}_{2}})}}$ of does not vanish at x(⁰), there exist neighborhoods of and on which f is bijective, i.e., the lens mapping is locally invertible. For an infinitesimal displacement dy of the source, the corresponding image position in the lens plane changes by $dx = {{A}^{{ - 1}}}({{x}^{{(0)}}})dy.$