The following is a paraphrased version of the derivation within John D. Anderson's Fundamentals of Aerodynamics's section on the Method of Characteristics:
The exact governing equation for two-dimensional supersonic, steady, and inviscid flow is as follows: \begin{equation} \left[ 1 - \frac{1}{a^2} \left( \phi_x \right)^2 \right] \phi_{xx} + \left[ 1 - \frac{1}{a^2} \left( \phi_y \right)^2 \right] \phi_{yy} = \frac{2}{a^2} \phi_x \phi_y \phi_{xy} \end{equation} where $a$ is the Speed of Sound, $\phi$ is the Velocity Potential, and subscripts represent partial derivatives with respect to the subscript variable. Since $ \phi_x \equiv u $ and $ \phi_y \equiv v $, the $x$- and $y$-components of the velocity field $\vec{u} = \langle u, v \rangle$, Eq. 1 can be rewritten as the following: \begin{equation} \left[ 1 - \frac{u^2}{a^2} \right] \phi_{xx} + \left[ 1 - \frac{v^2}{a^2} \right] \phi_{yy} = \frac{2uv}{a^2} \phi_{xy} \end{equation} By expanding the exact differentials for $ \phi_x $ and $ \phi_y $ as functions of $x$ and $y$, i.e. $ \phi_x = f(x, y) $, we can obtain the following: \begin{equation} \mathrm{d} \phi_x = \mathrm{d}u = \phi_{xx} \, \mathrm{d}x + \phi_{xy} \, \mathrm{d}y \end{equation} \begin{equation} \mathrm{d} \phi_y = \mathrm{d}v = \phi_{yy} \, \mathrm{d}y + \phi_{xy} \, \mathrm{d}x \end{equation} Therefore, Eqs. (2)-(4) can be consolidated as: \begin{equation} \begin{bmatrix} 1 - u^2/a^2 & 1 - v^2/a^2 & - 2uv/a^2 \\ \mathrm{d}x & 0 & \mathrm{d}y \\ 0 & \mathrm{d}y & \mathrm{d}x \end{bmatrix} \begin{bmatrix} \phi_{xx} \\ \phi_{yy} \\ \phi_{xy} \end{bmatrix} = \begin{bmatrix} 0 \\ \mathrm{d}u \\ \mathrm{d}v \end{bmatrix} \end{equation} Through the use of Cramer's Rule, we can obtain a solution for $\phi_{xy}$: \begin{equation} \phi_{xy} = \frac{ \left| \begin{matrix} 1 - u^2/a^2 & 1 - v^2/a^2 & 0 \\ \mathrm{d}x & 0 & \mathrm{d}u \\ 0 & \mathrm{d}y & \mathrm{d}v \end{matrix} \right| } { \left| \begin{matrix} 1 - u^2/a^2 & 1 - v^2/a^2 & - 2uv/a^2 \\ \mathrm{d}x & 0 & \mathrm{d}y \\ 0 & \mathrm{d}y & \mathrm{d}x \end{matrix} \right|} = \frac{N}{D} \end{equation} Evaluating the numerator and denominator determinants: \begin{equation} \begin{aligned} N & = \left[ 1 - \frac{u^2}{a^2} \right] \left| \begin{matrix} 0 & \mathrm{d}u \\ \mathrm{d}y & \mathrm{d}v \end{matrix} \right| - \left[ 1 - \frac{v^2}{a^2} \right] \left| \begin{matrix} \mathrm{d}x & \mathrm{d}u \\ 0 & \mathrm{d}v \end{matrix} \right| \\ & = - \mathrm{d}y \, \mathrm{d}u \left[ 1 - \frac{u^2}{a^2} \right] + \mathrm{d}x \, \mathrm{d}v \left[ 1 - \frac{v^2}{a^2} \right] \end{aligned} \end{equation} \begin{equation} \begin{aligned} D & = \left[ 1 - \frac{u^2}{a^2} \right] \left| \begin{matrix} 0 & \mathrm{d}y \\ \mathrm{d}y & \mathrm{d}x \end{matrix} \right| - \left[ 1 - \frac{v^2}{a^2} \right] \left| \begin{matrix} \mathrm{d}x & \mathrm{d}y \\ 0 & \mathrm{d}x \end{matrix} \right| \\ & \quad + \left[- \frac{2uv}{a^2} \right] \left| \begin{matrix} \mathrm{d}x & 0 \\ 0 & \mathrm{d}y \end{matrix} \right| \\ & = - \mathrm{d}y^2 \left[ 1 - \frac{u^2}{a^2} \right] + \mathrm{d}x^2 \left[ 1 - \frac{v^2}{a^2} \right] - \mathrm{d}x \, \mathrm{d}y \left[ \frac{2uv}{a^2} \right] \end{aligned} \end{equation}
\begin{equation} \phi_{xy} = \frac{- \mathrm{d}y \, \mathrm{d}u \left[ 1 - \frac{u^2}{a^2} \right] + \mathrm{d}x \, \mathrm{d}v \left[ 1 - \frac{v^2}{a^2} \right]}{- \mathrm{d}y^2 \left[ 1 - \frac{u^2}{a^2} \right] + \mathrm{d}x^2 \left[ 1 - \frac{v^2}{a^2} \right] - \mathrm{d}x \, \mathrm{d}y \left[ \frac{2uv}{a^2} \right]} \end{equation} One can see that for certain values of $\mathrm{d}x$ and $\mathrm{d}y$, $\phi_{xy}$ can be indeterminate.
My question: What would an indeterminate form in the derivative of a flow variable mean physically? As far as I am aware, this is not representative of a shock.