Time evolution of integrals

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

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

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.

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 ${\displaystyle \int _{\Omega }F\,d\Omega }$

is governed by the following law:[1]

${\displaystyle {\frac {d}{dt}}\int _{\Omega }F\,d\Omega =\int _{\Omega }{\frac {\partial F}{\partial t}}\,d\Omega +\int _{S}CF\,dS}$

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.

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

${\displaystyle \int _{S}F\,dS}$

The law reads

${\displaystyle {\frac {d}{dt}}\int _{S}F\,dS=\int _{S}{\frac {\delta F}{\delta t}}\,dS-\int _{S}CB_{\alpha }^{\alpha }F\,dS}$

where the ${\displaystyle {\delta }/{\delta }t}$-derivative is the fundamental operator in the calculus of moving surfaces, originally proposed by Jacques Hadamard. ${\displaystyle B_{\alpha }^{\alpha }}$ 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 ${\displaystyle B_{\alpha }^{\alpha }}$. 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 ${\displaystyle F\equiv 1}$ since ${\displaystyle \int _{S}\,dS}$ is area:

${\displaystyle {\frac {d}{dt}}\int _{S}\,dS=-\int _{S}CB_{\alpha }^{\alpha }\,dS}$

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

${\displaystyle C\equiv B_{\alpha }^{\alpha }}$

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

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

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:

${\displaystyle \int _{S}F\,dS}$

is

${\displaystyle {\frac {d}{dt}}\int _{S}F\,dS=\int _{S}{\frac {\delta F}{\delta t}}\,dS-\int _{S}CB_{\alpha }^{\alpha }F\,dS+\int _{\gamma }c\,d\gamma }$

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