Question

Consider F and C below.

F(x, y, z) = (2xz + y2)i + 2xy j + (x2 + 9z2)k
C: x = t2, y = t + 2, z = 2t - 1, 0 ≤ t ≤ 1

(a) Find a function f such that F = ∇f.


(b) Use part (a) to evaluate ∫C ∇f · dr along the given curve C.

Answer 1

The question asks for the determination of a potential function f from a given vector field F and to use that to evaluate the line integral of ∇f along a curve C. The potential function f is found by integrating the components of F, and the integral along C can be evaluated using the gradient theorem.

Step-by-step explanation:

The student's question involves finding a potential function f such that F = ∇f and using that to evaluate the integral of ∇f ⋅ dr along a curve C. First, we seek a function f so that its gradient is equal to the given vector field F. Once f is found, we can evaluate the integral using the gradient theorem which states that the integral of a gradient field along a curve from point A to point B is simply f(B) - f(A).

To find the potential function f, we integrate each component of F with respect to its corresponding variable, making sure that terms only involving other variables show up in all relevant components. If a potential function exists, these terms will align themselves and reveal the form of f.

Once f is found, we can parameterize the curve C using the given parametric equations and directly evaluate f at the endpoints of the parameter interval to find the value of the integral, since the integral of a gradient field over a curve depends only on the values of f at the boundaries of the curve.