Correct way to write multiple argument functions

[Stephen Tashi]

You mean that since L is uniquely determined, i.e. single valued, with respect to x, it should be a regular rather than a partial derivative and that is the bad notation?

Yes, that's what I mean, but we really shouldn't call the function "L". The physics approach doesn't introduce a new name for the function created by substituting functions of x for the arguments of L.

Think about what we teach students in secondary school about functions. A function has a domain and co-domain. If two functions have a different domain, they are not the same function. If "L" was the name of a unique function, it would have a unique domain. In the case at hand, the domain of L is triples of real numbers. When we create a different function whose domain is single real numbers, we shouldn't call that function "L" - without apologizing to our classes of secondary school students!

You may find notation like $\frac{dL}{dx}$ used to indicate a "total derivative" of the unnamed function. That notation may be used alongside the notation $\frac{\partial L}{\partial x}$ , which indicates a partial derivative of $L$ with respect to its first argument.