Total derivative (along a path)

Definition · applies to functions along paths · also called material derivative (in fluid mechanics)

Undergraduate

For a function $f(t, x)$ evaluated along a path $x(t)$, the total derivative is

\[\frac{\mathrm{d}}{\mathrm{d}t} f(t, x(t)) = \frac{\partial f}{\partial t} + \nabla_x f \cdot \dot{x}.\]

Notes

The total derivative accounts for the change in $f$ both from its explicit dependence on $t$ and through $x(t)$.

Another common meaning of “total derivative” is the linear map $Df(x)$: see total derivative (linear map) and the meanings of “total derivative”.

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