Let $\rho(x, y, z, t)$ and $u(x, y, z, t)$ represent density and velocity, respectively, at a point $(x, y, z)$ and time $t$. Assume $\frac{\partial \rho }{\partial t}$ is continuous. Let $V$ be an arbitrary volume in space enclosed by the closed surface $S$ and $\hat{n}$ be the outward unit normal of $S$. Which of the following equations is/are equivalent to $\frac{\partial \rho }{\partial t} + \nabla \cdot(\rho u) = 0$?
$\int\limits_{V} \frac{\partial \rho}{\partial t} \, dv = - \int\limits_{V} \nabla \cdot(\rho u) \, dv$
$\int\limits_{V} \frac{\partial \rho}{\partial t} \, dv = \int\limits_{V} \nabla \cdot(\rho u) \, dv$
Let $\mathbb{R}$ and $\mathbb{R}^3$ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of $\alpha$ for which the set of vectors
$$ \{ [2 \ -3 \ \alpha], \ [3 \ -1 \ 3], \ [1 \ -5 \ 7] \}$$
does not form a basis of $\mathbb{R}^3$ is _______.
Suppose $X$ and $Y$ are independent and identically distributed random variables that are distributed uniformly in the interval $[0,1]$. The probability that $X \geq Y$ is _______ .
Consider the Earth to be a perfect sphere of radius $R$. Then the surface area of the region, enclosed by the 60°N latitude circle, that contains the north pole in its interior is _______.