I have to show that if one of the following statements is true, then the other statement is also true.
Let f be a scalar field and let P be a point in the domain of f.
There is some Cartesian system in which f is represented by a function which has a linear approximation at P.
For every choice of Cartesian system, f is represented by a function which has linear approximation at P.
I don't really know what either statement means by itself, so it is difficult to show how one implies the other. In particular, what does it mean to choose a different Cartesian system (that sounds like just moving the origin around). Further, I don't know what the book is saying in an earlier paragraph when they say "the gradient vector can be visualized without reference to a particular coordinate system"
The only thing that I do know is that to have a linear approximation at a point means that the function is both continuous at the point and has partial derivatives at the point (there might be cases where this is not enough to say that a linear approximation exists however).
Thanks for helpful responses