# Time evolution of integrals

In many applications, one needs to calculate the rate of change of a volume or surface integral whose domain of integration, as well as the integrand, are functions of a particular parameter. In physical applications, that parameter is frequently time t.

## Introduction

The rate of change of one-dimensional integrals with sufficiently smooth integrands, is governed by this extension of the fundamental theorem of calculus:

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

The calculus of moving surfaces[1] provides analogous formulas for volume integrals over Euclidean domains, and surface integrals over differential geometry of surfaces, curved surfaces, including integrals over curved surfaces with moving contour boundaries.

## Volume integrals

Let t be a time-like parameter and consider a time-dependent domain Ω with a smooth surface boundary S. Let F be a time-dependent invariant field defined in the interior of Ω. Then the rate of change of the integral $\int_\Omega F \, d\Omega$

is governed by the following law:[1]

$\frac{d}{dt} \int_\Omega F \, d\Omega =\int_\Omega \frac{\partial F}{\partial t} \, d\Omega + \int_S CF \, dS$

where C is the velocity of the interface. The velocity of the interface C is the fundamental concept in the calculus of moving surfaces. In the above equation, C must be expressed with respect to the exterior normal. This law can be considered as the generalization of the fundamental theorem of calculus.

## Surface integrals

A related law governs the rate of change of the surface integral

$\int_S F \, dS$

$\frac{d}{dt } \int_S F \, dS = \int_S \frac{\delta F}{\delta t} \, dS - \int_S CB^\alpha_\alpha \, dS$

where the ${\delta}/{\delta} t$-derivative is the fundamental operator in the calculus of moving surfaces, originally proposed by Jacques Hadamard. $B^\alpha _\alpha$ is the trace of the mean curvature tensor. In this law, C need not be expression with respect to the exterior normal, as long as the choice of the normal is consistent for C and $B^\alpha_\alpha$. The first term in the above equation captures the rate of change in F while the second corrects for expanding or shrinking area. The fact that mean curvature represents the rate of change in area follows from applying the above equation to $F\equiv 1$ since $\int_S \, dS$ is area:

$\frac{d}{dt} \int_S S \, dS = -\int_S CB^\alpha_\alpha \, dS$

The above equation shows that mean curvature $B^\alpha_\alpha$ can be appropriately called the shape gradient of area. An evolution governed by

$C\equiv B^\alpha_\alpha$

is the popular mean curvature flow and represents steepest descent with respect to area. Note that for a sphere of radius R, $B^\alpha_\alpha = -2/R$, and for a circle of radius R, $B^\alpha_\alpha = -1/R$ with respect to the exterior normal.

## Surface integrals with moving contour boundaries

Illustration for the law for surface integrals with a moving contour. Change in area comes from two sources: expansion by curvature $CB^\alpha_\alpha dt$ and expansion by annexation $cdt$.

Suppose that S is a moving surface with a moving contour γ. Suppose that the velocity of the contour γ with respect to S is c. Then the rate of change of the time dependent integral:

$\int_S F \, dS$

is

$\frac{d}{dt} \int_S F \, dS = \int_S \frac{\delta F}{\delta t} \, dS - \int_S CB_\alpha^\alpha F \, dS + \int_\gamma c \, d\gamma$

The last term captures the change in area due to annexation, as the figure on the right illustrates.

## References

1. ^ a b Grinfeld, P. (2010). "Hamiltonian Dynamic Equations for Fluid Films". Studies in Applied Mathematics. doi:10.1111/j.1467-9590.2010.00485.x. ISSN 00222526.