Time evolution of integrals

Short description: Change of time of the value of an integral

Within differential calculus, 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.

## Contents

* 1 Introduction
* 2 Volume integrals
* 3 Surface integrals
* 4 Surface integrals with moving contour boundaries
* 5 References

## Introduction

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

[math]\displaystyle{ \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) }[/math]

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 [math]\displaystyle{ \int_\Omega F \, d\Omega }[/math]

is governed by the following law:[1]

[math]\displaystyle{ \frac{d}{dt} \int_\Omega F \, d\Omega =\int_\Omega \frac{\partial F}{\partial t} \, d\Omega + \int_S CF \, dS }[/math]

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

[math]\displaystyle{ \int_S F \, dS }[/math]

The law reads

[math]\displaystyle{ \frac{d}{dt } \int_S F \, dS = \int_S \frac{\delta F}{\delta t} \, dS - \int_S CB^\alpha_\alpha F \, dS }[/math]

where the [math]\displaystyle{ {\delta}/{\delta} t }[/math]-derivative is the fundamental operator in the calculus of moving surfaces, originally proposed by Jacques Hadamard. [math]\displaystyle{ B^\alpha _\alpha }[/math] 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 [math]\displaystyle{ B^\alpha_\alpha }[/math]. 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 [math]\displaystyle{ F\equiv 1 }[/math] since [math]\displaystyle{ \int_S \, dS }[/math] is area:

[math]\displaystyle{ \frac{d}{dt} \int_S \, dS = -\int_S CB^\alpha_\alpha \, dS }[/math]

The above equation shows that mean curvature [math]\displaystyle{ B^\alpha_\alpha }[/math] can be appropriately called the shape gradient of area. An evolution governed by

[math]\displaystyle{ C\equiv B^\alpha_\alpha }[/math]

is the popular mean curvature flow and represents steepest descent with respect to area. Note that for a sphere of radius R, [math]\displaystyle{ B^\alpha_\alpha = -2/R }[/math], and for a circle of radius R, [math]\displaystyle{ B^\alpha_\alpha = -1/R }[/math] 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 [math]\displaystyle{ CB^\alpha_\alpha dt }[/math] and expansion by annexation [math]\displaystyle{ cdt }[/math].

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:

[math]\displaystyle{ \int_S F \, dS }[/math]

[math]\displaystyle{ \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 }[/math]

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

## References

1. ↑ 1.0 1.1 Grinfeld, P. (2010). "Hamiltonian Dynamic Equations for Fluid Films". Studies in Applied Mathematics. doi:10.1111/j.1467-9590.2010.00485.x. ISSN 0022-2526.

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