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?

  • 4
    $\begingroup$ 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


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.


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:

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


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

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

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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