1.1. Multivariable Function

To be short, the multivariable function is a function that depends on several arguments.

A real function with n variables has a domain as a subset of $\mathbb{R}^{n}$, a range as a subset of $\mathbb{R}$. On 3-dimensional cartesian coordinates, the domain is a subset of xy plane and the range is a subset of all real numbers on z axis.

 

Similary, the graph of real function with n variables f is a set of points $(x_{1},x_{2},...,x_{n},f)$ where  $(x_{1},x_{2},...,x_{n})$ belongs to the domain D and $y=f(x_{1},x_{2},...,x_{n})$ belongs to the range R.

 

Considering the one-to-one match between point $(x_{1},x_{2},...,x_{n},f)$ in $\mathbb{R}^{n}$ and vector $\mathbf{x}=\left \langle x_{1},x_{2},...,x_{n},f \right \rangle$, real function with n variables f can be viewed as 3 different operations below.

1. Function of n real variables $x_{1},x_{2},...,x_{n}$

2. Function of a point variable $(x_{1},x_{2},...,x_{n})$

3. Function of a vector variable $\mathbf{x}=\left \langle x_{1},x_{2},...,x_{n},f \right \rangle$

 

As the most abstract form, real function with n variables f can be expressed as:

$$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$$