Let$\vec{F}=\left(3 x^{2}+y^{4} \sin x\right) \vec{i}+\left(4 y^{3} \cos x+z\right) \vec{j}-y \vec{k}$ Is the value of the line integral of $\vec{F}$ along any loop zero?
A)Yes; the function is a gradient vector field.
B)No; the function is not a gradient vector field.

Question

Quizplus · Accepted Answer

For a vector field $\vec{F}$ to be a gradient vector field, it must be conservative, meaning it has a potential function from which it is derived. A necessary condition for this is that the curl of $\vec{F}$ must be zero. Calculating the curl of $\vec{F}$ and finding it to be non-zero would indicate that $\vec{F}$ is not a gradient vector field, and thus the line integral of $\vec{F}$ along any loop would not necessarily be zero.

Let F⃗=(3x2+y4sin⁡x)i⃗+(4y3cos⁡x+z)j⃗−yk⃗\vec{F}=\left(3 x^{2}+y^{4} \sin x\right) \vec{i}+\left(4 y^{3} \cos x+z\right) \vec{j}-y \vec{k}F=(3x2+y4sinx)i+(4y3cosx+z)j​−yk

Let $\vec{F}=\left(3 x^{2}+y^{4} \sin x\right) \vec{i}+\left(4 y^{3} \cos x+z\right) \vec{j}-y \vec{k}$