EC > GATE 2024 > Vector Calculus
Let $ρ(x,y,z,t)$ and $u(x,y,z,t)$ represent density and velocity, respectively, at a point $(x,y,z)$ and time t.
Assume $\frac{∂ρ}{∂t}$ is continuous. Let V be an arbitrary volume in space enclosed by the closed surface S and $n̂$ be the outward unit normal of S.
Which of the following equations is/are equivalent to $\frac{∂ρ}{∂t}+∇⋅(ρu)=0$?
A
$\int_{V}\frac{∂ρ}{∂t}dv=-\iint_{S}ρu⋅n̂ ds$
B
$\int_{V}\frac{∂ρ}{∂t}dv=[\oint_{S}ρu⋅n̂ ds$
C
$\int_{v}\frac{∂ρ}{∂t}dv=-\int_{V}∇⋅(ρu)dv$
D
$\int_{V}\frac{∂ρ}{∂t}dv=\int_{V}∇⋅(ρu)dv$

Correct : A;C

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