Answer to Question #120948 in Calculus for Olivia

Question

Let F(x,y,z)=⟨2xy,x^2+z,y⟩ be a conservative vector field. A potential function of F

is
Select one:
a. x^2y+yz+z


b. x^2yz


c. x^2y+yz−87

d. x^6yz


e. ⟨x^2y,yz,z⟩

Answer 1

$F(x,y,z)=⟨2xy,x^2+z,y⟩$

We know that $F$ is the gradient of potential function $U(x,y,z).$ So

$\dfrac{\partial U}{\partial x} = 2xy \Rightarrow U(x,y,z) = x^2y + f_1(y,z), \\ \dfrac{\partial U}{\partial y} =x^2+z \Rightarrow U(x,y,z) = x^2y+yz+f_2(x,z), \\ \dfrac{\partial U}{\partial z} = y \Rightarrow U(x,y,z) =yz+f_3(x,y).$

So $U(x,y,x) = x^2y+yz+\mathrm{const}.$ Therefore, we should choose answer c. $U(x,y,x) = x^2y+yz-87.$