Multivariable functions

We already know how a function takes the value of an independent variable and outputs another value dependent on it. However, real-life phenomena can rarely be described using a function that depends on only one variable. For example, position in space is expressed using three variables (length, width, height), or the demand for a product depends on the price of goods and the income of the consumer (two variables). It can get even more complex: image recognition software might use each pixel from a 30x30 image of a handwritten character as input to produce the actual character as output ($30\cdot30$ = 900 variables!).

In this topic, we will take a look at the definition of a multivariable function, some related terminology, and a few examples.

Multivariable functions

Let $V$ and $W$ be two vector spaces, each containing a finite number of elements. A relation between them can be expressed by the mapping $T$ such that:$$T:V \to W$$$T$ associates every element of $V$ to an element of $W$. Also, let's say that $V$ is a set of real vectors with $n$ components and $W$ a set of real vectors with $m$ components. Then, we have:

$$T: \mathbb{R}^n \to \mathbb{R}^m$$

Now, if $T$ associates every element of $V$ to a unique element of $W$, we can call it a function. The relation becomes a function with domain in n-space $\mathbb{R}^n$ and with range in m-space $\mathbb{R}^m$:

$$f: Df \subseteq \mathbb{R}^n \to Rf \subseteq \mathbb{R}^m$$

We are already familiar with functions in which both $n$ and $m$ are equal to 1:

This type of function is called a real-valued function of a real variable. Functions like $f(x) = 3x$, $g(x) = x^2$, $h(x) = \sqrt{\frac{x+1}{5x}}$ are all real-valued functions of the real variable $x$.

When $n \geq 1$ and $m \geq 1$, we can say that we have a multivariable function:

Scalar field

When $m=1$, the function is a real-valued function of a vector variable or, as it is usually called, a scalar field:

For example, in cartography, topographic elevation maps show information about the height of the terrain above sea level; given two coordinates $x$ and $y$, we get another real number $z$ representing the terrain elevation: $z=f(x,y)$

Vector field

When $m \gt 1$, the function is a vector-valued function of a vector variable or, as it is usually called, a vector field.

For example, wind maps can be constructed by assigning a vector representing the wind speed and direction to each point on the map; given two coordinates $x$ and $y$, we get a vector $\vec{z}$ representing the wind velocity vector at that point: $\vec{z} = \begin{pmatrix} z_1 \\ z_2 \end{pmatrix} = f(x,y)$

There are many other examples of vector and scalar fields that can be visualized:

• Temperature shown in infrared cameras;

• Electric field and voltage from an electrical distribution cable.

And countless others that are more difficult to do so — it's hard for the human brain to comprehend more than three or four dimensions visually. Nonetheless, we can see how multivariable functions can be used to give a more accurate description of the world around us.

What about distances in multidimensional space?

Since multivariable functions are generalizations for n-dimensional (and m-dimensional) Euclidean spaces, we can also apply the generalized notion of "length" given by the Euclidean norm to the input and output variables, as they represent the n-tuples (and m-tuples) in these spaces:

$$\|\vec{x}\| = \|\vec{x}\|_2 = \sqrt{\langle \vec{x},\vec{x} \rangle} = \sqrt{x_1^2+x_2^2+ \cdots + x_n^2} \qquad ; \quad \vec{x}=(x_1, x_2, \dots x_n)$$

Conclusion

To sum up, in this topic we have learned that:

• Multivariable functions are similar to regular functions, except that instead of only one input and output, they can take any number of inputs and produce any number of outputs.
• When the output of a multivariable function is only one real number, we call it a scalar field. When the output is a vector of numbers, we call it a vector field.
• Since we are dealing with n-dimensional vectors, we can use generalizations like the Euclidean norm to find their length in n-dimensional space.