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:

ddt∫a(t)b(t)f(t,x)dx=∫a(t)b(t)∂f(t,x)∂tdx+f(t,b(t))b′(t)−f(t,a(t))a′(t)

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 ∫ΩFdΩ

is governed by the following law:[1]

ddt∫ΩFdΩ=∫Ω∂F∂tdΩ+∫SCFdS

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

∫SFdS

The law reads

ddt∫SFdS=∫SδFδtdS−∫SCBααFdS

where the δ/δt-derivative is the fundamental operator in the calculus of moving surfaces, originally proposed by Jacques Hadamard. Bαα 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 Bαα. 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≡1 since ∫SdS is area:

ddt∫SdS=−∫SCBααdS

The above equation shows that mean curvature Bαα can be appropriately called the shape gradient of area. An evolution governed by

C≡Bαα

is the popular mean curvature flow and represents steepest descent with respect to area. Note that for a sphere of radius R, Bαα=−2/R, and for a circle of radius R, Bαα=−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 CBααdt 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:

∫SFdS

ddt∫SFdS=∫SδFδtdS−∫SCBααFdS+∫γcdγ

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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