Physics I / Line Integrals
Practice question · Multiple choice

You must compute the work CFdr\int_C \vec{F}\cdot d\vec{r} along a complicated spiral path, and you notice F=f\vec{F} = \nabla f for some potential ff. What is the fastest correct move?

Hints
  1. The fundamental theorem for line integrals: for gradient fields, only the endpoints matter.
  2. Zero work is guaranteed only for closed loops. This path has distinct endpoints.
Show the answer

C. Evaluate ff at the endpoints: work =f(end)f(start)= f(\text{end}) - f(\text{start})

Why

Conservative fields make the path irrelevant: Cfdr=f(B)f(A)\int_C \nabla f \cdot d\vec{r} = f(B) - f(A) exactly, spiral or not. This is why potential energy exists, and why "work done against gravity" depends on height gained, never on the route taken.

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