A particle moves along the path from the origin to (0, 2, 1) to (1, 2, 1) to (1, 0, 0) and back to the origin under the influence of the force field F = z2i + 2xyj+ 4y2k.

Calculate the work done by direct calculation and by using Stokes’ Theorem

To calculate the work done by a force field, we can integrate the dot product of the force field and the path of the particle. Let's consider both methods:

1. Direct Calculation:
The dot product between the force field F = z^2i + 2xyj + 4y^2k and the path vector ∆r = dxi + dyj + dzk is given by F · ∆r = (z^2∆x) + (2xy∆y) + (4y^2∆z), where ∆x, ∆y, and ∆z are the changes in x, y, and z coordinates along the path.

Now, let's calculate the work done for each part of the path:

a) From the origin (0,0,0) to (0,2,1):
∆x = 0, ∆y = 2, ∆z = 1.
So, F · ∆r = (0^2 * 0) + (2 * 0 * 2) + (4 * 2^2 * 1) = 16.

b) From (0,2,1) to (1,2,1):
∆x = 1, ∆y = 0, ∆z = 0.
So, F · ∆r = (1^2 * 1) + (2 * 1 * 0) + (4 * 0^2 * 0) = 1.

c) From (1,2,1) to (1,0,0):
∆x = 0, ∆y = -2, ∆z = -1.
So, F · ∆r = (1^2 * 0) + (2 * 1 * -2) + (4 * (-2)^2 * -1) = -16.

d) From (1,0,0) back to the origin (0,0,0):
∆x = -1, ∆y = 0, ∆z = 0.
So, F · ∆r = (0^2 * -1) + (2 * 0 * 0) + (4 * 0^2 * 0) = 0.

The total work done is the sum of the work done along each part of the path:
Work = 16 + 1 - 16 + 0 = 1.

2. Using Stokes’ Theorem:
Stokes' Theorem relates the work done by a force field along a closed loop to the curl of the force field over the surface enclosed by the loop.

In this case, the loop is formed by the path of the particle. We need to find the curl of the force field.
Let F = z^2i + 2xyj + 4y^2k. The curl of F, denoted as ∇ × F, can be found by taking the determinant:

∇ × F = ∂(4y^2)/∂y - ∂(2xy)/∂z i - (∂(4y^2)/∂x - ∂(z^2)/∂z)j + (∂(2xy)/∂x - ∂(2xy)/∂y)k

Simplifying this, we get:
∇ × F = -2xi + 2yj + 0k

Now, the work done can be calculated as the surface integral of ∇ × F over the surface enclosed by the loop. Since the loop lies in the xy-plane, the z-component of ∇ × F is 0, which means the work done is 0.

Therefore, using Stokes' Theorem, the work done is 0.

Hence, the work done by direct calculation is 1, while using Stokes' Theorem, it is 0.