Curves and vectorial functions

Why stick to the simple "$y$ equals" when you can spice things up with a "$t$ for time" and see your equations come alive? Parametric equations are akin to giving x and y a pair of roller skates; suddenly, they're traveling swiftly, forming amazing paths.

In this topic, you'll get to know parametric curves better and start understanding their behavior. With this knowledge, you'll be prepared to expand the concept of the derivative to functions in higher dimensions. This notion is crucial in fields like machine learning, physics, economics, and engineering.

Curves

Now it's time to take on something you've already encountered: a cool-looking curve $C$ that isn't necessarily the graph of a function:

This means the curve $C$ cannot appear as a set of points $(x, f(x))$ for any given function $f$. So the strategies we've developed until now won't work. Does this mean you should give up on studying general curves? Even simple things like the unit circle, easily outlined by the points satisfying $x^2 + y^2 = 1$?

To address this, let's apply some physics and common sense. You can effortlessly describe a curve as the route taken by a moving particle in a plane. At a particular moment $t$, the particle will be at a specific point with two coordinates. The crucial idea here is to highlight the dependence of these coordinates on time $t$ by expressing them through two different functions, $u(t)$ and $v(t)$. Essentially, the particle's movement gives rise to two separate functions.

Conversely, if you have two functions, $u$ and $v$, you can construct a curve $C$ by plotting every point $(u(t), v(t))$ as $t$ varies. Consequently, we say that $C$ is parametrically represented by $u$ and $v$. Additionally, these two functions are referred to as the parametric representation of the curve.

With this fresh concept, the graph of any standard function $f$ becomes a simple curve. It's straightforward to describe it parametrically as $x=t$ and $y=f(t)$. Now let's get out there and graph some curves!

Plotting curves

Let's begin with a basic curve described by:

Do you recall your first experiences graphing functions? You likely created a table with numbers from the domain and their corresponding function values. You can do the same here, taking some values of $t$ between $-1$ and $2$ and calculating the values of $x$ and $y$.

These are some values of the curve. For a more detailed visual, plot the $(x, y)$ pairs and connect them:

The first instance is when $t=-1$, so the curve begins at the point $(0,-3)$. Likewise, at the final instance $t=2$, the curve ends at $(-3, 0)$. As time progresses, the curve moves from its starting point to its ending point in a determined direction.

In general, the curve with parametric equations $x=u(t)$ and $y=v(t)$, where $a \leq t \leq b$, has an initial point at $(u(a), v(a))$ and ends at the terminal point $(u(b), v(b))$. Before moving onto more theoretical concepts, let's examine a few more interesting curves.

Engaging examples

Trigonometric functions provide us the flexibility to produce more intriguing curves. Consider this critical example:

When you plot several points, a noticeable pattern emerges:

It appears to form a circle! In actuality, that's exactly what it is. As you add more points, the curve becomes clearer. Can you recall that the equation for the unit circle is $x^2 + y^2 = 1$? Well, the parametric equations meet this condition:

Here, time is considered as the angle, increasing from $0$ to $2 \pi$, so the point moves from the starting point $(0,1)$ around the circle in a counterclockwise direction to its terminal point $(0,1)$. This is an instance of a closed curve, where the starting and terminal points are the same.

Remember the curve we discussed at the beginning? Feel free to check out its parametric equations:

By blending trigonometric functions, you can create remarkably aesthetic and harmonious curves. Here are some stunning examples:

Now, fasten your seat belt because you're about to plunge deep into the world of mathematics.

Vectorial functions

Defining a curve in the plane through two functions might seem complex, but you can simplify this by redefining the original concept of a function.

Instead of a rule linking one number to another, you can consider a function $\mathbf{c}$ from real numbers to the plane. In simpler terms, a rule $\mathbf{c}$ that associates each number $t$ with a point on the plane, which you can represent as $\mathbf{c}(t)$.

With this new concept, a curve is merely a function that maps an interval of real numbers into the plane. Quite elegant, isn't it?

Just like you can add and multiply regular functions, you should also be able to do the same with this new type of functions. This capability will become very useful when we expand the concept of derivatives. Linear algebra will serve as the bridge, simplifying the handling of this new category of functions just as with ordinary ones.

Remember that points on the plane are referred to as vectors originating from $(0, 0)$, and they can be added and multiplied by scalars. This allows you to add points on the plane and also multiply them by numbers. That's why this new kind of function $\mathbf{c}$ is named a vectorial function. For each $t$, $\mathbf{c}(t)$ is a vector consisting of two coordinates, $u(t)$ and $v(t)$. Therefore, from this point on, the two definitions of curve you've built are essentially the same.

Now, let's assume we have two vector functions, $\mathbf{c}$ and $\mathbf{d}$, defined as $\mathbf{c}(t) =(u(t), v(t))$ and $\mathbf{d} (t) =(w(t), z(t))$, and a number $a$. Their sum and scalar product can be defined as:

That's more than sufficient for now. In the next topic, you'll use all this theory to build the derivative of vector functions. Get ready for some fun!

Conclusion

• A curve, denoted as $C$, is parametrically represented by $u$ and $v$. It comprises all the pairs $(x, y)$ where $x=u(t)$ and $y=v(t)$. These two equations are known as the parametric equations of the curve.

• Typically, $t$ falls within some interval $[a, b]$.

• The curve starts at the point $(u(a), v(a))$ and ends at the point $(u(b), v(b))$.

• To plot the curve, the most effective strategy is to create a table containing several values for $t$ along with their corresponding $x$ and $y$ values.

• The parametric equations for a unit circle are:

  \[x = \cos(t) \qquad y = \sin(t) \qquad 0 \leq t \leq 2 \pi\