These appear to hang on the definition of "smoothness" for a curve or a surface, and how coordinates are defined for either. He says: "... we may want to consider a quantity $$ \Phi\; =\; f(x,y) $$, defined on the surface, but expressed with respect to various different coordinate systems. The mathematical expression for the function $$ f(x,y) $$ may well change from patch to patch, even though the value of $$ \Phi $$ at any specific point of the surface 'covered' by those patches does not change."

He also describes partial differentiation of a function like $$ f(x,y) $$ as having one variable held constant. This must mean that x and y are treated as being independent variables in f. Does this mean that x and y are independent, or is that only the case when a differential operator is applied to one of them?

Another point he makes is that $$\frac {\partial} {\partial x} $$ and $$\frac {\partial} {\partial y} $$ can be interpreted as 'arrows' pointing along coordinate lines. This suggests that a system of coordinates is not necessarily a collection of points, yes? Differential operators as arrows also suggests direction vectors. Multiplying a partial differential operator by a real (or complex) number gives you a vector.

So what's the second fundamental confusion of calculus? This has to do with transforming (the function f(x,y)) to another coordinate system (say, f(X,Y)) and failing to note that y = constant might not agree with Y = constant. That is, y and Y might have different gradients even though they overlap the same region of a surface.

I think Tach (not sure about Pete) is making these mistakes in the debate about Lorentz transformations and angles that are invariant (transform conformally). But that's another time and place (or coordinate system). Another point Penrose seems to be making is in respect of the importance of establishing a Cauchy-Riemann condition, or "structural integrity" perhaps, between the systems of coordinates.