Determine whether the line integral F*ds is path-independent?

Question

Let R be the unit cube defined by 0<=x<=1, 0<=y<=1, 0<=z<=1, and let F(x, y, z)=(x+yz, y+xz, z+xy). Determine whether the line integral F*ds is path-independent.

Answer 1

The line integral is path - independent if this integral taken over a closed path is zero. This is possible if

                                             curl F(x,y,z) = 0.

        curl F = (∂Fz/∂y  -  ∂Fy/∂z)i + (∂Fx/∂z - ∂Fz/∂x) j + (∂Fy/∂x - ∂Fx/∂y) k

You can make sure taht 

∂Fz/∂y = ∂Fy/∂z = x                ∂Fx/∂z = ∂Fz/∂x = y         ∂Fy/∂x = ∂Fx/∂y = z

Thus,  curl F = 0 and the integral is path - independent. This means the force F is conservative.

Typical examples are gravitational field near the earth surface and homogeneous electric field, such as in capasitors made of two parallel plates (if we ignore boundary effects). Work done by these forces on closed path is zero (it is defined by a line integral). That is why the change in potential energy of an object moving in  homogeneous fields from one point to another depends on difference of potentials (gravitational or electrical) at these points only, but doesn't depend on path of motion.

Answer 2

Since ∇XF = i(x-x) - j(y-y) + k(z-z) = 0, F is conservative. Therefore, the line integral F*ds is path-independent.

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Attn: Use 3 by 3 determinant to calculate ∇XF.