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Exercise 18

Avg difficulty 2.5

Let $\vF=\langle F_1,F_2,F_3\rangle$ , where $F_1= 3xy^2+z$ and $F_3 = yz^2+\cos(z)$ .

Give a component function $F_2$ such that $\vF$ is a gradient vector field.
Give a component function $F_2$ such that $\vF$ is not a gradient vector field.

Long, Keller & Schlicker, Active Calculus Multivariable, §13.5 Path-Independent Vector Fields and the Fundamental Theorem of Calculus for Line Integrals, Exercise 18

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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.