Explanation:
The vector field F is conservative on R^3 if and only if it satisfies the condition of being a curl-free vector field.
We can check if F is conservative by computing the curl of F.
curl(F) = (∂Q/∂y - ∂P/∂z)i + (∂R/∂z - ∂P/∂x)j + (∂P/∂y - ∂Q/∂x)k
where P = 3y + 4z, Q = 3x + z, and R = 4x + y.
After computing the partial derivatives, we get:
curl(F) = 0i + 0j + 0k
Since the curl of F is zero, we can conclude that F is conservative on R^3.
To find the potential function, we need to integrate each component of F with respect to its corresponding variable.
φ(x,y,z) = ∫Pdx = ∫(3y + 4z)dx = 3xy + 4xz + C1(y,z)
φ(x,y,z) = ∫Qdy = ∫(3x + z)dy = 3xy + yz + C2(x,z)
φ(x,y,z) = ∫Rdz = ∫(4x + y)dz = 4xz + yz + C3(x,y)
where C1(y,z), C2(x,z), and C3(x,y) are arbitrary functions of the variables not being integrated.
To find the potential function, we can equate the three expressions for φ and solve for the arbitrary constants.
3xy + 4xz + C1(y,z) = 3xy + yz + C2(x,z) = 4xz + yz + C3(x,y)
Solving for the arbitrary constants, we get:
C1(y,z) = yz + C
C2(x,z) = 4xz + C
C3(x,y) = 3xy + C
where C is an arbitrary constant.
Therefore, the potential function for F is:
φ(x,y,z) = 3xy + 4xz + yz + C