Question

A scalar potential φ has the gradient ∇φ = yz î + xz ĵ + xy k̂. The value of the integral ∫_C ∇φ · dr̄ on the curve C: r̄ = x î + y ĵ + z k̂, parameterized as x=t, y=t², z=3t², for 1 ≤ t ≤ 3, is __________.

Accepted Answer

The correct answer is 726 (± 0.1). ∇φ = yz î + xz ĵ + xy k̂ is the gradient of φ = xyz (check: ∂(xyz)/∂x=yz etc.). For a conservative field, the line integral equals φ(end) - φ(start). At t=1: (x,y,z)=(1,1,3), φ=3. At t=3: (3,9,27), φ=3×9×27=729. Integral = 729-3 = 726.

Explanation