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I have a scalar deflection potential (in the study of weak lensing) and in the book (Schneider, Kochanek and Wambsganss's Gravitational Lensing: Strong, Weak and Micro) I have the following passage:

If a source is much smaller than the angular scale on which the lens properties change, the lens mapping can be linearised locally. The distortion of images is then described by

$$ \vec{A}(\vec{\theta}) =\frac{\partial\vec{\beta}}{\partial\vec{\theta}} =\Big(\delta_{ij}-\frac{\partial^2\psi(\vec{\theta})}{\partial\theta_i\theta_j}\Big) =\begin{pmatrix} 1-\kappa-\gamma_1 &\gamma_2\\ \gamma & -\kappa+\gamma_1\end{pmatrix}$$

where we have introduced the components of the shear $\gamma=\gamma_1+i\gamma_2=|\gamma|e^{2i\phi}$, $$\gamma_1=\frac{1}{2}(\psi_{,11}-\psi_{,22}),$$ $$\gamma_2=\psi_{,12}.$$

So there it is in context. Could someone please explan the notation used in the scalar potentials on the last 2 lines?

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    $\begingroup$ It's a derivative with respect to the coordinates indicated after the comma. $\endgroup$
    – Javier
    Oct 19, 2016 at 11:43
  • $\begingroup$ Matrices using \begin{pmatrix} a & b \\ c & d \end{pmatrix}. $\endgroup$ Oct 19, 2016 at 12:25
  • $\begingroup$ Those are the $\partial^2\psi(\theta)/\partial\theta_i\partial\theta_j$. $\endgroup$ Jun 28, 2021 at 14:46

2 Answers 2

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It means a partial derivative, the same as it usually does. In this case, $\psi$ is as scalar field, so it has no indices of its own, so there is nothing to "sandwich" (your word) the comma against on the left side of the subscript.

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The double subscripts denote twice partial differentiation. $(~_{22} ~ or ~_{11}) for~ principal, ~(.._{12}) for~ shear $ directions.

In case of curvature substitute $\kappa$ for $\gamma$, put away the old $\psi$ for the time being, for a symbol I am used to.

$ \theta$ is the displacement, primes are with respect to arc lengths.

$\psi$ is the angle made by a geodesic to principal directions. $\kappa_n) $ represents normal curvature, $\tau)g$ $ represents geodesic torsion

Euler's Curvature relation:

$$\kappa_n=\kappa_1 \cos^2 \psi + \kappa_2 \sin^2 \psi $$ $$ \tau_g= (\kappa_1-\kappa_2)\sin \psi\cos \psi $$

$$\kappa=\kappa_n+i\tau_g=|\kappa|e^{2i\psi}$$

$$|\kappa|=\frac{1}{2}(\theta_{,11}-\theta_{,22})= \frac12 (\kappa_{n~max} -\kappa_{n~min})= \frac12 (\kappa_1 -\kappa_2) $$

$$=\text{Radius of Mohr Circle}=|\tau_g|=\theta_{,12}= \text{max. shear stress}$$

Note that these can be represented by Mohr's Circle of Curvature (as here) for Stress, Strain, Moment of Inertia or any other tensor dependent on two direction/variables in continuum theory.

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