Recall that a function $f$ from a set $U$ to a set $V$ is a rule that assigns, to each $x\in U$, **one and only one** element $y\in V$. We call $x$ the **independent variable** and $y$ the **dependent variable**. We express it by writing $y=f(x)$. The sets $U$ and $V$ are called the domain and co-domain of $f$, respectively. To mention that $f$ is a function with the domain $U$ and the co-domain $V$, we write $f:U\to V$. In the calculus of a single variable, $U$ and $V$ are subsets of $\mathbb{R}$. In this chapter, we deal with functions where $U$ is a subset of $\mathbb{R}^n$.

**Definition:** A function $f:U\to\mathbb{R}$ where $U\subseteq \mathbb{R}^n$ is a rule that assigns one and only one real number to each point ${\mathbf x}=(x_1,\cdots,x_n)$ of $U$.

In a concise fashion, we may also write $f:U\subseteq\mathbb{R}^n\to\mathbb{R}$. When $n>1$, the function $f$ is called a **real-valued function of a vector variable** or simply a **scalar field**. In most of the examples in this chapter, $n$ is 2 or 3. When $n=2$, the independent variables are often denoted by $x$ and $y$ (or sometimes by $x_1$ and $x_2$), the dependent variable by $z$, and we write $z=f(x,y)$. When $n=3$, the independent variables are denoted by $x$, $y$ and $z$ (or by $x_1$, $x_2$ and $x_3$) and the dependent variable by another letter from the end of the alphabets like $u$ or $w$. In the most general case, we write $y=f(x_1,x_2,\cdots,x_n)$ or $y=f({\mathbf x})$; here $y$ is the dependent variable.

#### Elementary examples of multivariable functions

#### Elementary examples of multivariable functions

Polynomials are the simplest type of functions. A polynomial function of two variables $x$ and $y$ is the sum of a finite number of terms $c x^m y^n$ (called monomials), where $m$ and $n$ are nonnegative integers and $c$ is a real number. The degree of the monomial $c x^m y^n$ is $m+n$ providing $c\neq 0$. The degree of the monomial of three variables $cx^m y^n z^p$ is $m+n+p$ provided $c\neq 0$. The degree of a polynomial is the highest degree of its constituting monomials. Hence, the function defined by

\[z=-4 x^3 y^2+4 x^4+3 x y^2+7,\]
is a polynomial of degree 5.

Rational functions are the second simplest type of functions. A rational function is the quotient of one polynomial by another. Therefore, the general form of a rational function of two variables is

\[R(x,y)=\frac{P(x,y)}{Q(x,y)}\]
where $P(x,y)$ and $Q(x,y)$ are polynomials.

The functions that are generated by a finite number of operations addition, subtraction, multiplication, division, and raising to a fractional power are called algebraic functions, for example,

\[z=\sqrt[5]{\frac{x^3-y}{x^2 y^2+y}}+\sqrt[3]{\frac{x+\sqrt{xy}}{x+y^3}}.\]

When the domain of a function $y=f(x_1,x_2,\cdots,x_n)$ is not specified explicitly, we assume its domain is the set of all possible points in $\mathbb{R}^n$ at which $f$ produces real values. This set is called the **natural domain** or simply the **domain** of the function. The set of all possible outputs of the function is called the **range** of the function.

### Composition of functions

If $g$ is a function of one variable and $f$ is a function of three variables, then the composition of $g$ and $f$, $g\circ f$, is the function of three variables defined by

\[g\circ f(x,y,z)=g(f(x,y,z)).\]
The domain of $g\circ f$ consists of all points $(x,y,z)$ in the domain of $f$ such that $f(x,y,z)$ is in the domain of $g$. The extension of this to functions of several variables is easy. Let $f$ be a function of $n$ variables and $g$ be a function of a single variable, then

\[g\circ f (x_1,\cdots,x_n)=g(f(x_1,\cdots,x_n))\]
and the domain of $g\circ f$ is the set of all points $(x_1,\cdots,x_n)$ in the domain of $f$ such that $f(x_1,\cdots,x_n)$ is in the domain of $g$.

In a similar way, if ${\mathbf r}:I\subseteq \mathbb{R}\to \mathbb{R}^n$ is a vector-valued function and $f:U\subseteq \mathbb{R}^n\to \mathbb{R}$, the composite function $f\circ{\mathbf r}$ is a function from $I\cap \{t|\ {\mathbf r}(t)\in U \}$ to $\mathbb{R}$, and is defined by $f\circ{\mathbf r}(t)=f({\mathbf r}(t))$.