Question: Show that if $$ varphi(t) = int_{C(t)} mathbf{a} cdot mathbf{d}x, $$ when $a(x, t)$ is any suitably smooth vector field, and $C(t)$ is a circuit

Show that if $$ \varphi(t) = \int_{C(t)} \mathbf{a} \cdot \mathbf{d}x, $$ when $a(x, t)$ is any suitably smooth vector field, and $C(t)$ is a circuit consisting of the same fluid particles as time proceeds, then \frac{d\varphi}{dt} = \int_{C(t)} \left[\frac{\partial \mathbf{a}}{\partial t} + (\mathbf{a} \cdot abla) \mathbf{u} ight] \cdot \mathbf{d}x

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