8. Time derivative of integrals over moving domains#
Some results about time derivatives of integrals over moving domains are collected here. These results are useful for writing balance equations of physical quantities in integral form over arbitrary domains, like:
integral form of balance equations in continuum mechanics, see Continuum Mechanics:Governing Equations:Integral balance equations for arbitrary domains
integral form of Maxwell’s equations governing classical electromagnetism, see Electromagnetism:Principles of Classical Electromagnetism:derivation of balance equations for arbitary volume, starting from equations for a control volume
Link to hand-written notes.
8.1. Volume density#
Reynolds transport theorem. Given a volume \(V(t)\) with boundary \(\partial V(t)\), whose points \(\vec{r} \in \partial V(t)\) have velocity \(\vec{v}_b\),
\[\dfrac{d}{dt} \int_{V(t)} f = \int_{V(t)} \dfrac{\partial f}{\partial t} + \oint_{\partial V(t)} f \vec{v}_b \cdot \hat{n} \ .\]
“Proof”
\[\begin{split}\begin{aligned}
\dfrac{d}{dt} \int_{v(t)} f(\vec{r}, t) \, dt
& = \lim_{\Delta t \rightarrow 0 } \frac{1}{\Delta t} \left[ \int_{v(t+\Delta t)} f(\vec{r}, t+\Delta t) - \int_{v(t)} f(\vec{r}, t) \right] = \\
& = \lim_{\Delta t \rightarrow 0 } \frac{1}{\Delta t} \left[ \int_{v(t)} \big\{ f(\vec{r}, t+\Delta t) - f(\vec{r}, t) \big\} + \int_{\Delta v(t;\Delta t)} f(\vec{r}, t) \right] = \\
& = \lim_{\Delta t \rightarrow 0 } \frac{1}{\Delta t} \left[ \int_{v(t)} \left\{ \Delta t \, \dfrac{ \partial f}{\partial t}( \vec{r}, t) + o(\Delta t) \right\} + \oint_{\partial v(t)} \left\{ \Delta t \, \vec{v}_b \cdot \hat{n} \, f(\vec{r}, t) + o(\Delta t) \right\} \right] = \\
& = \int_{v(t)} \dfrac{\partial f}{\partial t} + \oint_{\partial v(t)} f \vec{v}_b \cdot \hat{n} \ .
\end{aligned}\end{split}\]
8.2. Flux across a surface#
\[\frac{d}{dt} \int_{S(t)} \vec{f} \cdot \hat{n} = \int_{S(t)} \frac{\partial \vec{f}}{\partial t} \cdot \hat{n} + \int_{S(t)} \nabla \cdot \vec{f} \,\, \vec{v}_b \cdot \hat{n} - \int_{\partial S(t)} \vec{v}_b \times \vec{f} \cdot \hat{t} \]
“Proof”
\[\begin{split}\begin{aligned}
\dfrac{d}{dt} \int_{s(t)} \vec{f}(\vec{r}, t) \cdot \hat{n} \, dt
& = \lim_{\Delta t \rightarrow 0 } \frac{1}{\Delta t} \left[ \int_{s(t+\Delta t)} \vec{f}(\vec{r}, t+\Delta t) \cdot \hat{n}(\vec{r}, t+\Delta t) - \int_{s(t)} f(\vec{r}, t) \cdot \hat{n}(\vec{r}, t) \right] = \\
& = \lim_{\Delta t \rightarrow 0 } \frac{1}{\Delta t} \left[ \int_{s(t+\Delta t)} \vec{f}(\vec{r}, t+\Delta t) \cdot \hat{n}(\vec{r}, t+\Delta t) - \int_{s(t)} f(\vec{r}, t) \cdot \hat{n}(\vec{r}, t) \right. + \\
& \left. \qquad \qquad - \int_{s(t)} \vec{f}(\vec{r}, t + \Delta t) \cdot \hat{n}(\vec{r}, t) + \int_{s(t)} \vec{f}(\vec{r}, t + \Delta t) \cdot \hat{n}(\vec{r}, t) \right] = \\
& = \lim_{\Delta t \rightarrow 0} \dfrac{1}{\Delta t} \left[ \int_{s(t)} \Delta t \dfrac{\partial \vec{f}}{\partial t} \cdot \hat{n} + o(\Delta t) + \oint_{\partial \Delta v(t;\Delta t)} \vec{f} \cdot \hat{n} - \oint_{\partial s(t)} \Delta t \vec{f} \cdot \hat{t} \times \vec{v}_b \right]= \\
& = \int_{s(t)} \dfrac{\partial \vec{f}}{\partial t} \cdot \hat{n} + \int_{s} \nabla \cdot \vec{f} \, \vec{v}_b \cdot \hat{n} - \oint_{\partial s(t)} \vec{v}_b \times \vec{f} \cdot \hat{t} \ .
\end{aligned}\end{split}\]
having used
\[\oint_{\partial \Delta v(t;\Delta t)} \vec{f} \cdot \hat{n} = \int_{\Delta v} \nabla \cdot \vec{f} = \Delta t \int_{s} \nabla \cdot \vec{f} \, \hat{n} \cdot \vec{v}_b + o(\Delta t)\]
8.3. Work line integral along a line#
\[\frac{d}{dt} \int_{\ell(t)} \vec{f} \cdot \hat{t} = \int_{\ell(t)} \frac{\partial \vec{f}}{\partial t} \cdot \hat{t} + \int_{\ell(t)} \nabla \times \vec{f} \cdot \vec{v}_b \times \hat{t} + \vec{f}_B \cdot \vec{v}_B - \vec{f}_A \cdot \vec{v}_A \]
“Proof”
\[\begin{split}\begin{aligned}
\dfrac{d}{dt} \int_{\ell(t)} \vec{f}(\vec{r}, t) \cdot \hat{t}(\vec{r},t) \, dt
& = \lim_{\Delta t \rightarrow 0 } \frac{1}{\Delta t} \left[ \int_{\ell(t+\Delta t)} \vec{f}(\vec{r}, t+\Delta t) \cdot \hat{t}(\vec{r}, t+\Delta t) - \int_{\ell(t)} \vec{f}(\vec{r}, t) \cdot \hat{t}(\vec{r}, t) \right] = \\
& = \lim_{\Delta t \rightarrow 0 } \frac{1}{\Delta t} \left[ \int_{\ell(t+\Delta t)} \vec{f}(\vec{r}, t+\Delta t) \cdot \hat{t}(\vec{r}, t+\Delta t) - \int_{\ell(t)} f(\vec{r}, t) \cdot \hat{t}(\vec{r}, t) \right. + \\
& \left. \qquad \qquad - \int_{\ell(t)} \vec{f}(\vec{r}, t + \Delta t) \cdot \hat{t}(\vec{r}, t) + \int_{\ell(t)} \vec{f}(\vec{r}, t + \Delta t) \cdot \hat{t}(\vec{r}, t) \right] = \\
& = \lim_{\Delta t \rightarrow 0 } \frac{1}{\Delta t} \left[ \int_{\ell(t+\Delta t)} \Delta t \, \dfrac{\partial \vec{f}}{\partial t}(\vec{r}, t+\Delta t) \cdot \hat{t}(\vec{r}, t+\Delta t) + \right. \\
& \left. \qquad \qquad + \oint_{\partial \Delta s(t)} f(\vec{r}, t) \cdot \hat{t}(\vec{r}, t) - \Delta t \vec{f}_A \cdot \vec{v}_A + \Delta t \vec{f}_B \cdot \vec{v}_B + o(\Delta t) \right] = \\
& = \int_{\ell(t)} \frac{\partial \vec{f}}{\partial t} \cdot \hat{t} + \int_{\ell(t)} \nabla \times \vec{f} \cdot \vec{v}_b \times \hat{t} + \vec{f}_B \cdot \vec{v}_B - \vec{f}_A \cdot \vec{v}_A \ .
\end{aligned}\end{split}\]
having used
\[\oint_{\partial \Delta s(t;\Delta t)} \vec{f} \cdot \hat{t} = \int_{\Delta s} \hat{n} \cdot \nabla \times \vec{f} = \Delta t \int_{\ell} \nabla \times \vec{f} \cdot \hat{v}_b \times \vec{t} + o(\Delta t)\]