Note 1
Multivariable Functions and Continuity
Finally, we are going to discuss about multivariable calculus! Before we get too excited, we need to discuss some of the fundamental topics on multivariable functions and their limits, continuity, and differentiability. In this note, we are going to discuss fundamental and necessary topics before investigating more into other calculus topics.
1.1 Multivariable Functions and Sets
As always, let’s get started with definitions and common notations. Consider the following notation. \[ \mathbb {R}^n \coloneqq \{ (x_1, x_2, \ldots , x_n) \mid x_i \in \mathbb {R} \} \] As you can notice from the equation, \(\mathbb {R}^n\) represents an \(n\)-tuple composed of real numbers. Some examples include \(\mathbb {R}^2\) and \(\mathbb {R}^3\), which represents the coordinates in 2-space and 3-space respectively.
Given this notation, we could naturally define a multivariable function \(f(x_1, x_2, \ldots , x_n)\) as a function that maps the elements of \(D \subset \mathbb {R}^n\) to a single real number. Here, the domain \(D\) is represented as \(\dom (f)\) and the range is denoted as \(\ran (f)\).
If the domain is not explicitly stated for a function, we could just assume it to be the largest subset of \(\mathbb {R}^n\) such that the function \(f\) is defined.
An interesting question to consider is how do you graph functions like \(f(x, y)\). The answer to this question is pretty straightforward given that the graph of \(f(x)\) is defined as \(\{ (x, y) \mid y = f(x) \}\). For \(f(x, y)\), we could simply extend it as \(\{ (x, y, z) \mid z = f(x, y) \}\). For functions with more variables, it is impossible, or very difficult, to visually graph them, but there is a strategy to get a glimpse on their appearance. To get to the strategy, consider the following definitions.
Definition 1.1.1
Given a function \(f(x, y)\), the level curve of \(f\) of height \(k\) is defined as the set \(\{ (x, y) \mid f(x, y) = k \}\). The collection of different level curve of \(f\) is known as a contour plot, or contour map of \(f\).
Recall that when we wanted to see how paraboloids look like, we drew many traces of the surface. This is exactly what it is! With the same idea, we can get a glimpse of how a complex functions in 3-space look like. Similarly, we could do the same for functions in \(4\)-space. Although it is almost impossible and very difficult, we can use this method to approximate its appearance.
Continuing from the discussion on multivariable functions, let’s now turn our attention to their domains. As always, let’s get started with definitions.
Definition 1.1.2
Given a point \(P\) in 2-space or 3-space, the open ball or radius \(r\) centered at \(P\) is defined as the set of all points that lie inside the ball and denoted as \(B_r(P)\). \[ B_r (x_0, y_0) = \left \{ (x, y) \mid \sqrt {(x - x_0)^2 + (y - y_0)^2} < r \right \} \] Given \(D\), a subset of \(\mathbb {R}^2\) or \(\mathbb {R}^3\), a point \(P\) is an interior point of \(D\) and boundary point of \(D\) if there exists \(r > 0\) such that \(B_r(P) \subset D\) and if \(B_r(P)\) contains all points in and not in \(D\) for all \(r > 0\) respectively.
An intuitive way of understanding this considering \(D\) as an area and the interior points are points inside that shaded area and the boundary points are the points that are on the boundary of \(D\). Continuing, we can continue with the following definitions.
Definition 1.1.3
The interior of a set \(D\) in 2-space or 3-space is the set of all interior points and the boundary of \(D\) is the set of all boundary points of \(D\). An open set is a set that does not contain its boundary points and a closed set is a set that contains all its boundary points.
Some interesting fact that we can notice from these definitions is that a set \(D\) in \(2\)-space or \(3\)-space is open and closed at the same time if and only if \(D = \mathbb {R}^2\), \(D = \mathbb {R}^3\), or \(D = \emptyset \). This is because the three sets have empty boundary and it follows the definition of open and closed set. Such sets that are both open and closed are known as clopen set (really love the name!). Continuing with this definition, we could also define the following.
Definition 1.1.4
A set \(D\) in \(2\)-space or \(3\)-space is bounded if for some point \(P\) and \(r > 0\), \(D \subset B_r(P)\). Contrastingly, a set is unbounded if there is no such choice of \(P\) and \(r\). Moreover, a point \(P\) is an accumulation point of \(D\) if for all \(r > 0\), there exists a point \(Q \neq P\) such that \(Q \in D\) and \(Q \in B_r(P)\).
There were lots of definitions! I think it was very interesting to learn different ideas on sets because understanding the concepts, or definitions, in sets really widens the understanding of what it means to get arbitrarily close to a number and what limits are. This is it for this section, and in the next section, we will discuss more on limits and continuity of multivariable functions.