In weak gravitational lensing, how is the magnification of a source the sum of the magnifications of all its images? From Gravitational Lensing: Strong, Weak and Micro by Schneider, Kochanek and Wambsganss we are told that the images of a small source are magnified by $|\mu(\vec{\theta_i})|$. Then it follows that the total magnification of a source at position $\vec{\beta}$ in the source plane is $$\mu(\vec{\beta})=\sum_i|\mu(\vec{\theta_i})|.$$ However, I don't see how this works, since while the deflection angle is a linear sum of deflections from point lenses, the magnification, which is the determinant of the Jacobian, is not linear in the lenses. 
Can someone provide me with a derivation for the above equation? (This is not provided in the book.)
 A: $\vec{\theta}_i(\vec{\beta})$ is simply the position of the i-th image of $\vec{\beta}$ in the image plane. This image is relatively magnified in brightness which practically, for a near-point source, means simply that more light from it gets to us. For the individual images, the magnification (or growth in point-brightness) is given as $|\mu(\vec{\theta}_i)|$.
Imagine we now have two lensed images of a single object, both with a magnification equal to one. This means that in total over both images we have two times the light from the source as compared to the case of no lensing. It is then natural to define the "total magnification" of the object as equal to two because we get twice as much from the source. But the word "magnification" does not really have the meaning of "growth in size", more "growth in brightness".
Similarly, one can define the total magnification of a generally lensed source as the sum of the relative amounts of brightness we get from the source through all the lensed images. This gives us the "total magnification" $\mu(\vec{\beta})$ (or "the total relative amount of light from the source") as$$\mu(\vec{\beta}) = \sum |\mu(\vec{\theta}_i)|$$
This is essentially a definitory relation and cannot be elucidated much beyond the argument given above.

This is simply the convention in the lensing community, even though it might seem a little bit contrived. Then again, "magnification" for a point source does not make a lot of sense either way. The unspoken assumption in the usual lensing theory is that the sources are almost point-like but of non-zero size. If you consider this limit, a lot of the lensing theory becomes clearer.
