Let's say we have some time-independent scalar field [itex]\phi[/itex]. Obviously [tex]\phi\left(\mathbf{q}\right)-\phi\left(\mathbf{p}\right) = \int_{\gamma[\mathbf{p},\,\mathbf{q}]} \nabla\phi(\mathbf{x})\cdot d\mathbf{x}.[/tex](adsbygoogle = window.adsbygoogle || []).push({});

This is of course still true if the path [itex]\gamma[/itex] is the trajectory of a particle moving through space. But let's say we have a time-dependent field instead, with [itex]\gamma[/itex] still being the trajectory of the particle. Will

[tex]\phi(\mathbf{q},t_2)-\phi(\mathbf{p},t_1)=\int_{\gamma[\mathbf{p},\,\mathbf{q},t]} \nabla\phi(\mathbf{x} (t))\cdot d\mathbf{x}?[/tex]

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# Gradient theorem for time-dependent vector field

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