Saladsamurai said:
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 f(x_1,x_2,...,x_n) 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.
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.
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