Questions: Fundamental Theorem for Line Integrals

Question 1

A particle moves from point A to point B through a vector field F along three different paths. The work done is 7 J, 7 J, and 19 J respectively. What does this tell you about F?

• A. F is conservative — two of the three paths agree
• B. F is definitely not conservative, since at least one path gives a different value
• C. F might be conservative; more paths must be tested to be certain
• D. F is conservative only along the paths that give 7 J

Answer: (B)

For a conservative field, the line integral depends only on the endpoints — every path from A to B must give the same value. A single discrepancy (19 J ≠ 7 J) proves the field is not conservative. There is no threshold of 'most paths agree'; path independence is absolute.

Question 2

You want to compute the work done by Earth's gravitational field on a satellite as it moves from low orbit to a higher orbit via a complex spiraling maneuver. The most efficient calculation strategy is:

• A. Parametrize the spiral path and integrate F · dr along it
• B. Compute the average gravitational force and multiply by path length
• C. Use the potential energy difference U(r₂) − U(r₁), since gravity is a conservative field
• D. Apply the virial theorem to find the average kinetic energy along the path

Answer: (C)

Gravity is conservative (F = ∇f where f = −GMm/r), so the fundamental theorem applies: ∫_C F · dr = f(B) − f(A). The entire path integral collapses to a difference of potential values at the two orbital radii — the spiraling route in between is irrelevant. This is the whole power of the theorem: path-independence lets you avoid computing the integral at all.

Question 3

If F is a conservative vector field, then the work done by F on a particle that travels along any closed loop (returning to its starting point) is zero.

• T. True
• F. False

Answer: True

This follows directly from the fundamental theorem: ∫_C F · dr = f(B) − f(A). If A = B (a closed loop), then f(B) − f(A) = 0. This is the defining property of conservative fields, and it is physically meaningful — gravity and electrostatic forces are conservative, meaning no net work is done over a closed trajectory. Non-conservative forces like friction always do negative work on a closed loop.

Question 4

The fundamental theorem for line integrals states that any vector field can be integrated by evaluating its associated scalar function at the path's endpoints.

• T. True
• F. False

Answer: False

Only conservative fields — those that can be written as F = ∇f for some scalar potential f — admit this shortcut. A generic vector field has no potential function, so its line integral genuinely depends on every detail of the path. The key step before applying the theorem is always: verify that F is conservative (e.g., by checking that curl F = 0 in a simply connected domain).

Question 5

Explain the analogy between the single-variable fundamental theorem of calculus and the fundamental theorem for line integrals.

Model answer: In one variable, ∫_a^b F'(x) dx = F(b) − F(a): integrating a derivative over an interval gives the net change in the function. In multiple dimensions, if F = ∇f (F is the gradient of f), then ∫_C F · dr = f(B) − f(A): integrating a gradient field over a curve gives the net change in the potential function. The analogy is exact — the gradient is the multivariable generalization of the derivative, and in both cases the integral collapses to boundary values, making the interior path irrelevant.

The key is recognizing that ∇f is the multivariable counterpart of F' in one dimension. Both theorems express the same idea: if you integrate the 'rate of change' of a function, you get the total change. The multivariable case is more powerful because it handles arbitrary paths in space, not just intervals on a line — but only for fields that are gradients of some scalar function.