a) \(\frac{D\rho}{Dt}=\frac{d\rho}{dt}+u \frac{d\rho}{dx}+v\frac{d\rho}{dy}+w\frac{d\rho}{dz}\)
b) \(\frac{D\rho}{Dt}=\frac{\partial\rho}{\partial t}+u \frac{d\rho}{dy}+v\frac{d\rho}{dz}+w\frac{d\rho}{dx}\)
c) \(\frac{D\rho}{Dt}=\frac{\partial\rho}{\partial t}+u \frac{d\rho}{dy}+v\frac{d\rho}{dz}+w\frac{d\rho}{dx}\)
d) \(\frac{D\rho}{Dt}=\frac{\partial \rho}{\partial t}+u\frac{\partial \rho}{\partial x}+v\frac{\partial \rho}{\partial y}+w \frac{\partial \rho}{\partial z}\)

Answer: d
Explanation: As the location coordinates (x, y, z) vary with time,
\(\frac{D\rho}{Dt}=\frac{\partial\rho}{\partial t}+\frac{\partial\rho}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial \rho}{\partial y}\frac{\partial y}{\partial t}+\frac{\partial \rho}{\partial z}\frac{\partial z}{\partial t}\)
\(\frac{D\rho}{Dt}=\frac{\partial \rho}{\partial t}+u\frac{\partial \rho}{\partial x}+v\frac{\partial\rho}{\partial y}+\frac{\partial \rho}{\partial t}\)