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Up to date as of January 14, 2010

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The Leibniz rule

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 + Δa and b < x < b + Δb, 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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