Groundwater Contaminant Plume in Radial Aquifer (Environmental Engineering)

An environmental engineer is modeling the chemical contaminant concentration $C(x, y)$ (in milligrams per liter, $\text{mg/L}$ ) leaking from an underground storage tank into a uniform, porous sand aquifer:

C(x, y) = 100 e^{-0.02(x^2 + y^2)}

where $x$ and $y$ represent Cartesian coordinates in meters measured relative to the source leak.

To monitor groundwater remediation and design circular extraction well barriers, the engineering team transforms the analytical model from Cartesian coordinates to polar coordinates ( $r, \theta$ ) via:

x(r, \theta) = r\cos\theta, \qquad y(r, \theta) = r\sin\theta

State the Case 2 Multivariable Chain Rule formulas for $\frac{\partial C}{\partial r}$ and $\frac{\partial C}{\partial \theta}$ .
Compute the Cartesian partial derivatives $\frac{\partial C}{\partial x}$ and $\frac{\partial C}{\partial y}$ .
Evaluate the radial concentration gradient $\frac{\partial C}{\partial r}$ at a monitoring radius of $r = 10\text{ m}$ .
Show analytically that $\frac{\partial C}{\partial \theta} = 0$ , and interpret the environmental significance of this result for circular barrier placement.