Let,
t → Instantaneous time
\( \vec{v} \) → Velocity in x-direction
\( \vec{x} \) → Instantaneous position
\( \vec{x_0} \)→ Initial position
The relationship between Eulerian and Lagrangian approaches for velocity in x direction is given by _______
a) \( \vec{v}(t, \vec{x}(\vec{x_0}, t)) = \frac{\partial \vec{x} (\vec{x_0}, t)}{\partial t} \)
b) \( \vec{v}(t, \vec{x}(\vec{x_0}, t)) = \frac{\partial \vec{x_0} (\vec{x}, t)}{\partial t} \)
c) \( \vec{v}(t, \vec{x_0}(\vec{x}, t)) = \frac{\partial \vec{x} (\vec{x_0}, t)}{\partial t} \)
d) \( \vec{v}(t, \vec{x_0}(\vec{x}, t)) = \frac{\partial \vec{x_0} (\vec{x}, t)}{\partial t} \)

Answer: a
Explanation: Location at any instantaneous time is
\(\vec{x}(\vec{x_0}, t) \)
Velocity of fluid flow in Lagrangian approach is
\( \frac{\partial \vec{x}(\vec{x_0}, t)}{\partial t}\)
Velocity of fluid flow in Eulerian approach is
\( \vec{x}(t, \vec{x} (\vec{x_0}, t)) \)
The relationship between the approaches is
\(\vec{x}(t, \vec{x} (\vec{x_0}, t)) = \frac{\partial \vec{x}(\vec{x_0}, t)}{\partial t} \).