### Function notation

Recall the notation that $\R$ stands for the real numbers. Similarly, $\R^2$ is a two-dimensional vector, and $\R^3$ is a three-dimensional vector.

#### Scalar-valued functions

In one-variable calculus, you worked a lot with one-variable functions, i.e., functions from $\R$ onto $\R$. If $f(x)$ is such a one-variable functions, we can write $f: \R \to \R$ as a shorthand way of expressing that $f$ is a function from $\R$ onto $\R$.

A function like $f(x,y) = x+y$ is a function of two variables. It takes an element of $\R^2$, like $(2,1)$, and gives a value that is a real number (i.e., an element of $\R$), like $f(2,1)= 3$. Since $f$ maps $\R^2$ to $\R$, we write $f : \R^2 \to \R$. We can also use this “mapping” notation to define the actual function. We could define the above $f(x,y)$ by writing $f: (x,y) \mapsto x+y$.

To contrast a simple real number with a vector, we refer to the real
number as a *scalar*. Hence, we can refer to $f : \R^2 \to \R$
as a scalar-valued function of two variables or even just say it is a
real-valued function of two variables.

Everything works the same for scalar valued functions of three or more variables. For example, $f(x,y,z)$, which we can write $f : \R^3 \to \R$, is a scalar-valued function of three variables.

#### Vector-valued functions

In contrast, a *vector-valued function* takes on values that
are vectors. First, let's talk about vector-valued functions of a
single variable.

A vector-valued function in two dimensions can be written $f : \R \to \R^2$. An example is $\vc{f}(t)=(3t,-t)$. For a given real number, which we'll denote by $\clubsuit$ for fun, $\vc{f}(\clubsuit)$ is the two-dimensional vector $(3\clubsuit,-\clubsuit)$. Similarly, a vector-valued function in three dimensions can be written $f : \R \to \R^3$. For example, if $\vc{f}(s) =(1-s,s^3, \cos s)$, then $\vc{f}(0) = (1,0,1)$. We sometimes write vector-valued functions using the standard unit vector $\vc{i}$, $\vc{j}$, and $\vc{k}$, as in $\vc{f}(s) = (1-s)\vc{i} + s^3 \vc{j} + (\cos s) \vc{k}$.

Lastly, we can have vector-valued functions of multiple variables. For example, a function could take values in $\R^3$, say $(x,y,z)$, and map them to $\R^2$, such as $f(x,y,z) = (x-y, x^{22}/z)$. We can write a function from $\R^3$ to $\R^2$ as $f: \R^3 \to \R^2$. You get the idea.

#### The domain of a function

The function $f(t) = (t, t^2)$ is defined over all real numbers $\R$,
i.e., the *domain* of the function is $\R$. Sometimes a
function of one variable may be defined over a subset of real numbers,
say some set $U \subset \R$; in this case, the domain of the function is
$U$. (Note, the symbol “$\subset$” just means “is subset of”.) In three dimensions, for example, we can specify the domain by
writing $\vc{f} : U \subset \R \to \R^3$, or simply $\vc{f} :U \to
\vc{R}^3$.

Example: since $\log t$ isn't a real number for $t \le 0$, the domain of $\vc{f}(t) = (\log t) \vc{i} + t \vc{j}$, is the set $D=(0, \infty)$. We could write this $\vc{f} : (0,\infty) \to \R^2$. What would the domain be if we replaced $\log t$ with $\log (t-3)$ or $\log (2-t)$? You have to think where $\log (t-3)$ or $\log (2-t)$ is a real number, i.e., where $t-3>0$ or where $2-t>0$.

We use the same notation for functions of multiple variables. If we wrote $\vc{f}: U \subset \R^2 \to \R^3$, we would mean a function maps values in a subset $U$ of $\R^2$ to values in $\R^3$.

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