  We present a streamlined, slightly modified version, in the two-variable
situation, of a beautiful, but not so well known, theory by B\"{o}gel, already
from the 1930s, on an alternative higher dimensional calculus of real
functions, a double calculus, which includes two-variable extensions of many
classical results from single variable calculus, such as Rolle's theorem,
Lagrange's mean value theorem, Cauchy's mean value theorem, Fermat's extremum
theorem, the first derivative test, and the first and second fundamental
theorems of calculus.
