Another Memory of Karl Menger: Concerning the notation for partial derivatives
Felix Rosenthal
BS EE 1947, MS Mathematics 1948, PhD Mech E 1952
Several Remembrances of Professor Menger have to do with his concern about notations. I recall a particular such matter in which his view was so correct that I could never understand why it is not being applied by everyone.
Consider a function of the form f[(x, g(x, y)].
What here is meant by the partial derivative ∂f/∂x?
Is it meant to be just the derivative of f with respect to its first variable x holding g constant, or is it the entire partial derivative, including the variation of g with x and just holding y constant? Using the universally applied "∂" notation, there is no way to tell.
Menger suggested that we should instead use subscripts instead of the ambiguous ∂ notation. Thus, the derivative of f with respect to its first variable x would be denoted by
f1(x, g).
On the other hand, the second interpretation would be written instead as
f1(x, g) + f2(x, g) * g1(x, y).
With this notation there can be no ambiguity.