In a book it says that:
"f is a scalar function. Hence its value at a point P depends on P but NOT on the particular choice of coordinates."
I do not understand this statement. Its value depends on P but doesn't P depend on the choice of coordinates?
When the coordinate system changes, would not the coordinates of point P change? The function may change also. But I checked some simple examples, it did not work.
Could you please elaborate the statement?
Points (as vectors segments, straight lines ecc.) are geometric entities that do not depend from the coordinate system chosen to represent them as ''numerical'' entities.
Use pencil and paper and fix two points on the paper, these points define a segment and a vector if you give an orientation to this segment. All these entities are defined without coordinates.
If you want to make some calculus with these entities, then coordinates can be useful and, at this point you chose a coordinate system. This means that you fix a point on the paper used as origin, two straight and oriented lines ( that can be orthogonal) and two unit measures on these lines.
Now a point can be represented by two numbers that are its coordinate on the chosen coordinate system. But you can chose another coordinate system and you have, for the same point, different coordinates.
A scalar function is a function that has a value that is , in general, different for different points, but is the same if a given point is represented in different coordinate systems, i.e. a function whose value does not depend from a change of coordinates.