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I am performing numerical and uncertainty analysis for an oblique shockwave angle function:

$$ \tan(\delta)=\frac{2}{\tan(\theta)}\frac{M^2\sin^2(\theta)-1}{M^2(\gamma+\cos(2\theta))+2} $$

where $\delta$ is deflection angle, $\theta$ is shock angle, M is mach number, and $\gamma$ is the ratio of specific heats. In our observation, $\delta$, $\theta$, and $M$ all have uncertainty. $\gamma$ is assumed to be exact.

Our unknown quantity is $\theta \pm \sigma_{\theta}$. If it was possible to solve for $\theta$ analytically, it would be trivial to propagate uncertainty (excluding covariances):

$$ \sigma_{\theta}^2 = \left( \frac{\partial\theta}{\partial M}\sigma_M \right)^2+ \left( \frac{\partial\theta}{\partial\delta}\sigma_{\delta} \right)^2 $$

The question is: how do I define $\theta$ so that I can take a derivative of it with respect to $M$ and $\delta$ for uncertainty analysis? Or maybe there is a way to approximate $\theta$ and take a numerical derivative? I am probably overthinking this.

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I'll try to address the question in bold. The expression for the angle function can be expressed as $f(\delta, M; \theta)$. You can compute partial derivatives of this function as $$ \frac{\partial f}{\partial \delta} = \frac{\partial f}{\partial \theta} \frac{\partial \theta}{\partial \delta} $$ and $$ \frac{\partial f}{\partial M} = \frac{\partial f}{\partial \theta} \frac{\partial \theta}{\partial M} $$ That will give you the two partial derivatives of $\theta$ that you seek.

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