In calculus, it is well-established that the integral of a line (1-D domain) in 2-D space can be described according to this relation:
$$ \int_a^b F({\bf r}(t))\frac{\partial {\bf r}}{\partial t} dt $$
where the line is described parametrically by ${\bf r}(t)$ for $t$ between $a$ and $b$. $F$ is a scalar field in 2-D space.
It is also well-established that the integral of a surface (2-D domain) in 3-D space can be described according to this relation:
$$ \int_{a_1}^{b_1}\int_{a_2}^{b_2} F({\bf r} (t_1, t_2)) \left|\frac{\partial {\bf r}}{\partial t_1} \times \frac{\partial {\bf r}}{\partial t_2}\right| \partial t_2 \partial t_1 $$
where the line is described parametrically by ${\bf r}(t_1, t_2)$ for $t_1$ between $a_1$ and $b_1$ and $t_2$ between $a_2$ and $b_2$. $F$ is a scalar field in 3-D space.
I see the clear similarities between these two relations, it is an integral of the field over a parametrically-described domain. However, is there any general equation that describes the integral of an N-1 D domain in N space?
Do you just keep appending more partials to the $\left|\frac{\partial {\bf r}}{\partial t_1} \times \frac{\partial {\bf r}}{\partial t_2}\right| $ factor?