The work done by the conservative force field $\vec{F} = (2xy + z^3)\hat{i} + (x^2)\hat{j} + (3xz^2)\hat{k}$ in moving an object from point $(0, 0, 0)$ to point $(1, 2, 1)$ is ________ (answer in integer).
Answer Key:
3
Step-by-Step Solution & Explanation
Find the scalar potential function $\phi(x, y, z)$ such that $\vec{F} = \nabla \phi$:
$\frac{\partial \phi}{\partial x} = 2xy + z^3 \implies \phi = x^2 y + x z^3 + g(y, z)$
$\frac{\partial \phi}{\partial y} = x^2 + \frac{\partial g}{\partial y} = x^2 \implies \frac{\partial g}{\partial y} = 0$
$\frac{\partial \phi}{\partial z} = 3xz^2 + \frac{\partial g}{\partial z} = 3xz^2 \implies \frac{\partial g}{\partial z} = 0$
Thus, $\phi(x, y, z) = x^2 y + x z^3$.
By the fundamental theorem of line integrals, the work done is:
$W = \phi(1, 2, 1) - \phi(0, 0, 0) = [1^2(2) + 1(1)^3] - 0 = 2 + 1 = 3$