  Gauss' and Stokes' theorems are fundamental results in vector calculus and
important tools in physics and engineering. When students are asked to describe
the meaning of Gauss' divergence theorem, they often use statements like this:
"The sum of all sources of a vector field in a region gives the net flux out of
the region". In order to raise this description to a mathematically sound level
of understanding, we present an educational approach based on the visual
interpretation of the vector differential operators, i.e. divergence and curl.
As a starting point, we use simple vector field diagrams for a qualitative
approach to connect both sides of the integral theorems, and present an
interactive graphical tool to support this connection. The tool allows to
visualise two-dimensional vector fields, to specify vector decomposition, to
evaluate divergence and curl point wise, and to draw rectangles to determine
surface and line integrals. From a meta-perspective, we situate this
educational approach into learning with (multiple) representations. Based on
prior research, the graphical tool addresses various learning difficulties of
vector fields that are connected to divergence and curl. The tool was
incorporated into the weekly lecture-based recitations of Physics II
(electromagnetism) in 2022 and 2023, and we assessed various educational
outcome measures. The students overall reported the tool to be intuitive and
user-friendly (level of agreement $76\%$, $N=125$), considered it helpful for
understanding and recommended its use for introductory physics courses (level
of agreement $65\%$, $N=65$).
