Time evolution of integrals

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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.


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[edit]

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[edit]

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

 \int_S F \, dS

The law reads

 \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[edit]

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


 \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.


  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.