triple integral – Math Memo Pad

Let us begin with an example (adapted from Schey) as our motivation. Recall from elementary calculus that the mass $M$ of a volume $V$ can be expressed as the triple integral of the density function $\rho(x,y,z)$ over $V$ :

$M = \iiint\limits_V \, \rho(x,y,z) \, \mathrm{d}V$

If density varies with time, i.e. $\frac{\partial \rho}{\partial t} \neq 0$ , then $M$ will also vary with time:

$\frac{\partial M}{\partial t} = \frac{\partial}{\partial t}M = \frac{\partial}{\partial t} \iiint\limits_V \, \rho(x,y,z) \, \mathrm{d}V = \iiint\limits_V \, \frac{\partial \rho}{\partial t} \, \mathrm{d}V$

The $\frac{\partial}{\partial t}$ can be moved “inside” the integral sign thanks to the Leibniz Integration Rule (formula below from the Wikipedia page), which states that:

$\frac{d}{dx} \left(\int_{a}^{b} f(x,t)\,dt \right)= \int_{a}^{b}\frac{\partial}{\partial x} f(x,t) \,dt.$

Caveats:

According to Schey (p. 50), “to be able to move the derivative under the integral sign this way requires $\frac{\partial \rho}{\partial t}$ to be continuous.”

Applications:

Used to derive the continuity equation (Schey, p. 50.)

References:

Harry M Schey. Div, grad, curl, and all that: an informal text on vector calculus. WW Norton, 3 edition, 1997.
Differentiation Under the Integral Sign on Brilliant.
Leibniz Integral Rule on Wikipedia.

Tag: triple integral

Leibniz Integration Rule / Moving a derivative under the integral sign