Final answer:
In order to find a function f such that F = ∇f, take the partial derivatives of f and equate them to the components of F. The work done by F on a particle moving along the curve C can be computed by evaluating the line integral ∫F⋅dr over the curve C.
Step-by-step explanation:
In order to determine a function f such that F = ∇f, where F is a vector field, we can hypothesize a function f(x, y, z) = ax^2 + by^2 + cz^2, introducing constants a, b, and c.
Deriving the partial derivatives of f, we acquire ∂f/∂x = 2ax, ∂f/∂y = 2by, and ∂f/∂z = 2cz.
Setting these derivatives equal to the corresponding components of F, we get 2ax = x, 2by = y, and 2cz = z.
Solving for a, b, and c reveals a = 1/2, b = 1/2, and c = 1/2. Thus, f(x, y, z) = 1/2(x^2 + y^2 + z^2).
To compute the work done by F on a particle following curve C, we evaluate the line integral ∫F⋅dr over C.
Substituting C's parametric equations into F, we get F(t) = (1 + 4sint)i + (1 + 4sin^2t)j + (1 + sin^3t)k, where 0 ≤ t ≤ π/2. By calculating the dot product F⋅dr and integrating over C, we obtain the work.