This is a very basic question, but one I keep getting confused on nonetheless. Can someone explain to me the difference between a scalar function and a vector function? I've been told that the only difference is in the way the quantities are regarded (one as scalars, the other as elements of a vector space) but I'm still quite confused... the terms keep popping up in relativity and I'm not sure what distinction I'm supposed to be making.
A scalar is a function that it's value is independent from the change of coordinates, while a vector has compontents that do dependent from the coordinate system.
A scalar function maps something to a scalar; a vector function maps something to a vector. So, to distinguish the two you simply ask what does the function return, a scalar or a vector. Example: Let x and y be vectors. An example of a vector function of x would be f(x) = 5x. (f(x) returns a vector.) An example of a scalar function of x would be g(x) = x dot y (the dot product of x and y), a scalar.
The difference between a vector and a scalar is very obvious... but I have to admit, I thought it was peculiar that in my electromagnetism book, a great distinction is made between the use of a vector function for surface integrals versus a scalar function for volume integrals. I wondered what would require one to have an output in a vector space, the other to not... physically, if I'm calculating an electric field or something, why should the electric field be a vector field in one dimension and a scalar field in the other? But I like the response about coordinate change. That does make a lot of sense.
Could you be more specific? What exactly is the "electric field or something" that you are trying to calculate? Are you talking about Gauss's law relating the surface integral (of the electric field component normal to a surface) to the total charge within the volume enclosed by the surface? If so, both the volume integral over the charge density (a scalar function of the position vector x) and the surface integral of E dot n (a scalar function = the dot product of the electric field vector E(x) with the vector n normal to the surface) produce a scalar result; otherwise, they could not be equal! In the surface integral, you are making use of a vector function, E(x) (the electric field at position x) but it is dotted with vector n to give a scalar. In the volume integral you are making use of the charge density at position x, a scalar function. Have I understood you, or do you mean something else?
Basically a vector function would be as an example r(t)=ti+tj+(t^2)k which in this case is a parabola in R3. The position, direction and magnitude of r(t) changes as t increases. For a scalar function, only its magnitude changes as the independent variable t changes, i.e. f(t) is contained in R meaning that a scalar function has no direction.