Difference between Scalar Function and Vector Function?

Saladsamurai

Okay I know the definition of a Vector and of a scalar.... but I am getting a little confused for some reason.

Wolfram.com gives this definition of a scalar function:

A function [itex]f(x_1,x_2,...,x_n)[/itex] of one or more variables whose range is one-dimensional, as compared to a vector function, whose range is three-dimensional (or, in general, n-dimensional).

So if what if n=1 then is the vector function really a scalar function?

I know this is a stupid question, but if I can answer it, I may or may not have to ask a series of even more stupid questions.

tiny-tim

Hi Saladsamurai!

Short answer … yes!

(that's assuming you're referring to the second n )

Tac-Tics

Wolfram is usually not very good when it comes to general mathematics definitions, in my experience. Usually, Wikipedia has a leg up on, so you might want to check their page.

In general, each function is associated with two sets called the domain and codomain. A function is a way to take elements from the domain and associate them with elements of the codomain. A function f with domain A and codomain B is notated f : A -> B (read f is a function from set A to set B).

So, standard real-valued functions you study in algebra are would be notated R -> R. You can also use your knowledge of the function to be more specific. The exponential function exp(x) = e^x can take any real number as input, but always outputs a *positive* number, so we can say exp: R -> R+ (exp is a function from the reals to the positive reals). Similarly, sin(x) takes any number and maps it to a number between -1 and 1, so we can say sin: R->[-1,1] (sine is a function from the reals to the closed interval between -1 and 1).

When you get to about your third year of calculus, you start dealing with functions other than R->R. We start working with R^2, R^3, or in general, R^n. These are all just sets, of course.

A function like f(x, y) = x^2 + y^2 is a function of two real variables. But we often blur the distinction and pretend like it's a function of a single vector variable instead. The output, though, is a real number (because x and y are real, their squares are real, and their sums are real). So we would say f: R^2 -> R. This is often called a scalar function in physics.

A function f(x, y) = (x^2, y^2) is also a function of two real variables. But the output this time is a vector. So f: R^2 -> R^2. Because the output is a vector, physicists often call this a vector field or a vector function.

So summary: scalar-function and vector-function refer to the *codomain* of a function, and general, the domain is assumed to be R^2 or R^3.

A few concrete examples. Energy potential as a function of space is a scalar function. At each point in space (represented by a vector), there is a single energy potential (a scalar).

A force field is a vector function. At each point in space (represented by a vector), there is a force that applies to objects at that location (another vector).

Lastly, about your question where n=1. Consider this. R^3 is a 3-dimensional space. R^2 is a plane. What does R^1 look like? It looks like a line. But interestingly, it looks EXACTLY like R. The two are called "isomorphic" because for every point on R^1 there is exactly corresponding real number and vise-versa. Here, again, we blur the distinction, and many authors would just go ahead and say R = R^1, but it's all a matter of interpretation.

tiny-tim

Hi Tac-Tics!
i'm not an author, but i'd say it also …

how are you defining R^1?

Tac-Tics

In a very straightforward manner.

You take an n-tuple to be a function from the set 0, 1, 2, ..., n to some other set.

For example, R^n represents the set of functions from {1, 2, ..., n} to R. This leads to the same familiar R^n for n>= 2. But additionally, it produces two more sets R^1 and R^0.

R^1 is a set of functions and is distinct from R, but it is isomorphic to R. All elements in R^1 have the form f(0) = x (for some real x).

R^0 is a silly trivial space, consisting of a single function: the function which maps the empty set to the reals. (It's a vacuous existence, but an existence none the less!) The single function in R^0 is the zero vector in that space.

Of course, this is all technical nonsense ;-)