# 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.)

$\vec{\theta}_i(\vec{\beta})$ 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 $|\mu(\vec{\theta}_i)|$.
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.