Continuum mechanics/Leibniz formula

The integral

F(t) = \int_{a(t)}^{b(t)} f(x, t)~\text{dx}

is a function of the parameter $t$ . Show that the derivative of $F$ is given by

\cfrac{dF}{dt} = \cfrac{d}{dt}\left( \int_{a(t)}^{b(t)} f(x, t)~\text{dx} \right) = \int_{a(t)}^{b(t)} \frac{\partial f(x, t)}{\partial t}~\text{dx} + f[b(t),t]~\frac{\partial b(t)}{\partial t} - f[a(t),t]~\frac{\partial a(t)}{\partial t}~.

This relation is also known as the Leibniz rule.

Proof:

We have,

\cfrac{dF}{dt} = \lim_{\Delta t\rightarrow 0} \cfrac{F(t + \Delta t) - F(t)}{\Delta t} ~.

Now,

\begin{align} \cfrac{F(t + \Delta t) - F(t)}{\Delta t} & = \cfrac{1}{\Delta t} \left[ \int_{a(t+\Delta t)}^{b(t+\Delta t)} f(x, t+\Delta t)~\text{dx} - \int_{a(t)}^{b(t)} f(x, t)~\text{dx}\right] \\ & \equiv \cfrac{1}{\Delta t} \left[ \int_{a+\Delta a}^{b+\Delta b} f(x, t+\Delta t)~\text{dx} - \int_{a}^{b} f(x, t)~\text{dx}\right] \\ & = \cfrac{1}{\Delta t} \left[ -\int_{a}^{a+\Delta a} f(x, t+\Delta t)~\text{dx} + \int_{a}^{b+\Delta b} f(x, t+\Delta t)~\text{dx} - \int_{a}^{b} f(x, t)~\text{dx}\right] \\ & = \cfrac{1}{\Delta t} \left[ -\int_{a}^{a+\Delta a} f(x, t+\Delta t)~\text{dx} + \int_{a}^{b} f(x, t+\Delta t)~\text{dx} + \int_{b}^{b+\Delta b} f(x, t+\Delta t)~\text{dx} - \int_{a}^{b} f(x, t)~\text{dx}\right] \\ & = \int_{a}^{b} \cfrac{f(x, t+\Delta t) - f(x,t)}{\Delta t}~\text{dx} + \cfrac{1}{\Delta t}\int_{b}^{b+\Delta b} f(x, t+\Delta t)~\text{dx} - \cfrac{1}{\Delta t}\int_{a}^{a+\Delta a} f(x, t+\Delta t)~\text{dx} ~. \end{align}

Since $f(x,t)$ is essentially constant over the infinitesimal intervals $a < x < a+\Delta a$ and $b < x < b+\Delta b$ , we may write

\cfrac{F(t + \Delta t) - F(t)}{\Delta t} \approx \int_{a}^{b} \cfrac{f(x, t+\Delta t) - f(x,t)}{\Delta t}~\text{dx} + f(b, t+\Delta t)~\cfrac{\Delta b}{\Delta t} - f(a, t+\Delta t)~\cfrac{\Delta a}{\Delta t}~.

Taking the limit as $\Delta t\rightarrow 0$ , we get

\lim_{\Delta t \rightarrow 0} \left[\cfrac{F(t + \Delta t) - F(t)}{\Delta t}\right] = \lim_{\Delta t \rightarrow 0}\left[ \int_{a}^{b} \cfrac{f(x, t+\Delta t) - f(x,t)}{\Delta t}~\text{dx}\right] + \lim_{\Delta t \rightarrow 0}\left[f(b, t+\Delta t)~\cfrac{\Delta b}{\Delta t}\right] - \lim_{\Delta t \rightarrow 0}\left[f(a, t+\Delta t)~\cfrac{\Delta a}{\Delta t}\right]

or,

{ \cfrac{dF(t)}{dt} = \int_{a(t)}^{b(t)} \frac{\partial f(x, t)}{\partial t}~\text{dx} + f[b(t),t]~\frac{\partial b(t)}{\partial t} - f[a(t),t]~\frac{\partial a(t)}{\partial t}~. } \qquad\qquad\qquad\square

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