Automated Audio Amplifier Loop Gain (Electrical & Computer Engineering)

In professional audio electronics design, an automated gain control circuit stabilizes output volume using an active feedback amplifier loop. The loop amplification gain factor $G$ (dimensionless ratio) depends on an input control bias voltage $v$ (in Volts, $\text{V}$) and a digital potentiometer feedback resistance $r$ (in kilo-ohms, $\text{k}\Omega$, for $r > 0$):

1. Compute the first-order partial derivatives $G_v$ and $G_r$.
2. Compute the second-order partial derivatives ($G_{vv}, G_{rr}, G_{vr}, G_{rv}$).
3. Verify Clairaut's Theorem for the mixed partials $G_{vr}$ and $G_{rv}$.
4. Construct the $2 \times 2$ Hessian matrix $\mathbf{H}_G(v, r)$ and evaluate it at the reference bias operating point $(v, r) = (2\text{ V}, e\text{ k}\Omega)$.
5. Calculate the Hessian determinant $D = \det(\mathbf{H}_G)$ at $(2, e)$ and determine whether the gain surface exhibits a local extremum or a saddle behavior at this operating point.