Assume δxi, δzi << dx, dy that is δt is small. Approximate the area A'B'C'D' as (A'D') · (A'B'). Neglect non–linear terms such as δxi · δzi; and obtain continuity equation (conservation of mass) such that ∇ · V~ = 0, where V~ is the velocity field.

Basically question asks me to obtain conversation of mass equation in 2-dimensional cartesian coordinates ( (∂u/∂x) + (∂w/∂z) = 0 ) by proving the area of fluid particle ABCD equals to A'B'C'D'.
(initially fluid particle is assumed to be rectangle and area of the rectangle is length*height=dx*dz. After short time it becomes A'B'C'D' and its area should be equal to the initial area.)

To obtain the conservation of mass equation (∇ · V~ = 0), we need to prove that the area of the fluid particle ABCD is equal to the area of A'B'C'D', under the assumption that δxi, δzi << dx, dy.

1. Start by considering the fluid particle ABCD in a 2-dimensional Cartesian coordinate system. The coordinates of the four vertices of the particle ABCD are (x, z), (x + dx, z), (x + dx, z + dz), and (x, z + dz).

2. After a short time interval δt, the fluid particle moves and becomes A'B'C'D' with the coordinates (x + δx, z + δz), (x + dx + δx, z + δz), (x + dx + δx, z + dz + δz), and (x + δx, z + dz + δz).

3. Using the assumption that δxi, δzi << dx, dy, we can neglect the nonlinear terms such as δxi · δzi in the expressions for the new coordinates of A'B'C'D'.

4. The new coordinates of A'B'C'D' can be approximated as (x + δx, z + δz), (x + dx, z + δz), (x + dx, z + dz), and (x + δx, z + dz).

5. The lengths of the sides of ABCD and A'B'C'D' can be approximated as:
- Side AD: δx
- Side AB: dx
- Side BC: dz
- Side CD: δz

6. The area of ABCD is given by length * height, which is dx * dz.

7. The area of A'B'C'D' is given by length * height, which is (x + δx - x) * (z + dz - z) = δx * dz.

8. Equating the areas of ABCD and A'B'C'D', we have dx * dz = δx * dz.

9. Dividing both sides of the equation by δt, we get (dx * dz) / δt = (δx * dz) / δt.

10. In the limit as δt approaches zero, the ratio dx * dz / δt approaches the partial derivative (∂t(Area)) and the ratio δx * dz / δt approaches the partial derivative (∂t(A'B'C'D')). Therefore, the equation becomes (∂t(Area)) = (∂t(A'B'C'D')).

11. Since (∂t(Area)) is the time rate of change of the area of ABCD, it represents the change in mass enclosed by ABCD.

12. Similarly, (∂t(A'B'C'D')) is the time rate of change of the area of A'B'C'D', which represents the change in mass enclosed by A'B'C'D'.

13. The continuity equation (conservation of mass) in 2-dimensional Cartesian coordinates is given by (∂u/∂x) + (∂w/∂z) = 0, where u and w are the x- and z-components of the velocity field V~.

Thus, by considering the approximation of the fluid particle and comparing the areas, we have derived the continuity equation (∇ · V~ = 0) in 2-dimensional Cartesian coordinates.