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Exercise 21

Avg difficulty 2.5

Let $\vF(x,y,z)$ be a vector field such that $\vF=\nabla f$ for a smooth function $f(x,y,z)$ . Let $S$ be a level surface of $f$ and let $C_1$ be curve that stays on the surface $S$ . Let $C_2$ be a curve that begins and ends on $S$ but does have points not on $S$ .

What is the value of $\int_{C_1} \vF \cdot d\vr$ ?
What is the value of $\int_{C_2} \vF \cdot d\vr$ ?

Long, Keller & Schlicker, Active Calculus Multivariable, §13.5 Path-Independent Vector Fields and the Fundamental Theorem of Calculus for Line Integrals, Exercise 21

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Adapted from Active Calculus Multivariable © 2013–2026 Nicholas Long, Mitchel T. Keller, and Steven Schlicker. Licensed under CC BY-SA 4.0. Source: Active Calculus Multivariable. Imported at revision 692c03c.