Past exam of the mathematical methods II course of the University of Oxford / 2014 / 12

The components of the curl of a gradient are differences of mixed second derivatives:If $\varphi$ has continuous second partial derivatives in a neighborhood, equality of mixed derivatives makes every component vanish. ThusThe $C^2$ condition is a sufficient regularity hypothesis; no simple-connectivity assumption is needed for this local identity. Topology matters for the converse assertion that every curl-free field has a global potential.

For the given polynomial vector field, direct differentiation givesAn exact differential is a differential one-form equal to $d\varphi ={\varphi }_{x}dx+{\varphi }_{y}dy+{\varphi }_{z}dz$ for a single-valued scalar function. This field is smooth on the simply connected domain ${\mathbb{R}}^3$, so its vanishing curl is sufficient for exactness. More concretely,The explicit potential proves exactness globally, rather than relying on curl-free behavior without checking the domain.

Integrate the first component of $\mathbf{F}=\nabla \varphi$ with respect to $x$:Matching the $y$ component gives $g_y=0$, and matching the $z$ component gives $g_z=0$. HenceThe fundamental theorem for line integrals now yieldsThe potential is a single-valued polynomial everywhere, so any piecewise smooth path with these endpoints has the same integral. The plus-gradient convention here should not be confused with the minus gradient of a mechanical potential energy.

Parametrize any actual particle trajectory by time, with $\mathbf{v}=\dot{\mathbf{r}}$ and acceleration $\mathbf{F}=\dot{\mathbf{v}}$. The chain rule givesIntegrating from the origin to the endpoint givesHere $K$ is specific kinetic energy, because the source defines it without a mass factor. Equivalently, $|{\mathbf{v}}_A{|}^2-|{\mathbf{v}}_O{|}^2=2$ in the chosen units. The individual initial or final speeds are not fixed by this result. For a particle of mass $m$, the change of ordinary kinetic energy would be $mI$.