# Intuition about moment function derivation [OR] derivatives involving a time varying integration domain

$$m_{{pq}}(t)=\iint\limits_{R(t)}h(x,y) dx dy$$ where $R(t)$ the domain of integration is time varying (In fact it is the only one which is time varying). And $$h(x,y) = x^p y^q f(x,y) dx dy$$

In general, $f(x,y)$ is a 2 dimensional function projected onto the power basis. To provide more context, just imagine that $f(x,y)$ is a binary image with only 1 white object of some random shape having a contour C(t). With respect to time, the white object displaces itself in the image and thus the meaning of time varying domain. The function $f(x,y)$ does not vary with respect to time. (The intensities of the object int the image are going to be constant).

If I need to differentiate the first equation $m_{{pq}}(t)$ to get the derivative w.r.t time, we can write
$$$$\dot{m}_{pq}=\oint\limits_{C(t)} h(x,y)\; \dot{\textbf{x}}^T \textbf{n}\,dl \label{eq:momentderiv}$$$$

where
$\dot{\textbf{x}}$ is the velocity of the point $\textbf{x}$ in the x,y image plane
$\textit{dl}$ is the infinitesimal increment along contour C(t)
$\textbf{n}$ is the unit normal vector to the contour at any point $\textbf{x}$ on it.

I am able to interpret the dot product between $\dot{\textbf{x}}$ and $\textbf{n}$ as the area of the infinitesimal surface between the contours at $C(t)$ and $C(t+dt)$. What i am not able to see is how exactly can the second equation be formally derived from the first equation?

Further, by using the Green's theorem, the second equation (derivative of moment) above can be written in terms of divergence as $$$$\dot{m}_{pq}= \iint\limits_{R(t)}\, \,div(h(x,y)\cdot\dot{\textbf{x}}) \,dx\,dy \label{eq:momentgreen}$$$$ I thought i had grasped the Green's theorem well but i dont see how this equation was obtained from the earlier moment derivative?

I did a formal course before many years before without actually understanding it thoroughly.I have been quite out of touch with multivariable calculus for long.To be comfortable with interpreting and solving these kind of equations, what readings do you suggest? Books? articles? I wish to develop intuition as well as mathematical techniques to solve these kind of problems that often appear in my domain.

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