What's the relationship between Gauss' law and Newton-Leibniz formula? Actually it's a puzzle I got in my Physics class.
Someone says Gauss' law actually is a specific example of the famous Newton-Leibniz formula, but I couldn't catch it.
So far I haven't learned about multivariable calculus, so maybe I need a simpler explanation~
Remark: Gauss' law: 
a link
 A: The question seems to refer to a connection between (part of) the fundamental theorem of calculus and Stokes' theorem in $n$-dimensional space.
If the real valued function $f(x)$ has an antiderivative $F$ on $[a,b]$, that is $f(x) = F'(x)$, then
$\int_a^b f(x) dx = F(b)-F(a).$
More suggestively,
\begin{equation*}
\int_a^b F'(x) d x = F(b) - F(a),\tag{1}
\end{equation*}
the integral of the derivative over the interval can be found knowing only the value of the function at the boundary of the interval.
Stokes' theorem is a generalization,
\begin{equation*}
\int_M d\omega = \int_{\partial M} \omega.\tag{2}
\end{equation*}
This theorem says that we can find the integral of the exterior derivative of the form $\omega$ over the $n$-dimensional manifold $M$ by integrating the form over the boundary $\partial M$ of the manifold.
If you don't know what a form is, never mind. It is our $F$.
And $d$ is our derivative.
And $M$ is our volume.
Equation (2) is totally analogous to (1), we are finding the sum of the derivative of some object over a volume knowing only the value of that object on the boundary.
An appropriate response to this result is amazement.
When we say that Stokes' theorem is a generalization of (1) we mean that for $n=1$ it is (1).
(1) is a special case.
Another special case of Stokes' theorem is the divergence theorem in three-dimensional space,
$$\oint_V  \nabla\cdot {\bf F}\, dV = \oint_{\partial V}  {\bf F}\cdot d{\bf S}.$$
If you continue on in physics you will learn very well this and other related results in multidimensional calculus in a good course on electromagnetism.
Rather than say more about the divergence theorem, we give below a summary of the important analogies between the various theorems.
$$\begin{array}{llcccccc}
&
\textrm{Theorem}
    & \textrm{Dimension}
    & \textrm{Object}
    & \textrm{Derivative}
        & \textrm{Volume}
            & \textrm{Surface}  \\ \hline
\textrm{Stokes'}
    & \int_M d\omega = \int_{\partial M} \omega
    & n
    & \omega
    & d\omega
        & M
            & \partial M \\
 \\
\textrm{Fund. Thrm.}
    & \int_a^b F'(x) d x = F(b) - F(a)
    & 1
    & F(x)
    & F'(x)
        & [a,b]
            & a,b \\
  \\
\textrm{Divergence}
    & \oint_V  \nabla\cdot {\bf F}\, dV = \oint_{\partial V}  {\bf F}\cdot d{\bf S}
    & 3
    & {\bf F}
    & \nabla\cdot {\bf F}
        & V
            & \partial V \\
\end{array}$$ 
A: It's not a "specific example" but rather can be seen as an extension to higher dimensions. The Newton-Leibniz formula gives the integral of a function derivative in terms of the value of the function on the "surface" of a line (i.e the end points). 
Similarly, the Divergence Theorem (as Gauss's Law is known in mathematics), expresses the integral over the divergence of the electric field which is the current density (loosely a 3D derivative) in terms of the value of the electric field on the surface of the volume. 
