The Euler's Equation of Motion
The Euler's equation of motion is a fundamental equation that describes the motion of an inviscid fluid. It is derived from the Navier-Stokes equations, which are a set of equations that govern the motion of a fluid. The Euler's equation of motion is a simplified form of the Navier-Stokes equations that neglects the effects of viscosity.

Conservation of Momentum for an Inviscid Fluid
The correct answer to the question is option B, which states that the Euler's equation of motion is a statement of conservation of momentum for an inviscid fluid. This means that the equation describes how the momentum of the fluid is conserved as it flows.

In fluid mechanics, momentum is a fundamental concept that describes the motion of a fluid. It is defined as the product of the mass and velocity of the fluid. Conservation of momentum means that the total momentum of a system remains constant unless acted upon by an external force.

Derivation of the Euler's Equation of Motion
The Euler's equation of motion can be derived by considering the conservation of momentum for a fluid element. Let's assume a small volume of fluid, called a control volume, with dimensions dx, dy, and dz. The control volume is moving with a velocity (u, v, w) in the x, y, and z directions, respectively.

The equation of motion can be written as:
ρ * (du/dt) = -∂P/∂x + ρ * gx
ρ * (dv/dt) = -∂P/∂y + ρ * gy
ρ * (dw/dt) = -∂P/∂z + ρ * gz

where ρ is the density of the fluid, du/dt, dv/dt, and dw/dt are the accelerations in the x, y, and z directions, respectively, P is the pressure, and gx, gy, and gz are the components of the gravitational acceleration in the x, y, and z directions, respectively.

Simplification for an Inviscid Fluid
In the case of an inviscid fluid, the effects of viscosity can be neglected. This means that the term on the left-hand side of the equation, which represents the acceleration, can be set to zero. The equation then becomes:

-∂P/∂x + ρ * gx = 0
-∂P/∂y + ρ * gy = 0
-∂P/∂z + ρ * gz = 0

These equations are known as the Euler's equations of motion. They describe the balance between pressure forces and gravitational forces in an inviscid fluid. The equations can be further simplified by assuming that the fluid is incompressible, which means that its density is constant. This leads to the following form of the Euler's equations:

∂P/∂x = -ρ * gx
∂P/∂y = -ρ * gy
∂P/∂z = -ρ * gz

Conclusion
In conclusion, the Euler's equation of motion is a statement of conservation of momentum for an inviscid fluid. It describes the balance between pressure forces and gravitational forces in the fluid. The equation is derived from the Navier-Stokes equations