Vector valued functions are usually encountered within machine learning, computer graphics, and computer vision algorithms. They are specifically useful for defining the parametric equations of space curves. It is critical to obtain a fundamental understanding of vector valued functions to understand more complicated concepts.

In this tutorial, you will find out what vector valued functions are, how to go about defining them and some instances.

After finishing this guide, you will be aware of:

• The definition of vector valued functions
• Derivatives of vector valued functions

Tutorial summarization

This guideline is subdivided into two portions, they are:

1. Definition and instances of vector valued functions
2. Differentiating vector valued functions

Definition of a vector valued function

A vector valued function is also referred to as a vector function. It is a function with the following two attributes:

1. The domain is a group of real numbers
2. The range is a group of vectors

Vector functions, are, thus, merely an extension of scalar functions, where both the domain and the range are the group of real numbers.

In this guide, we’ll take up vector functions whose range is the group of two or three dimensional vectors. Therefore, such functions can be leveraged to define a group of points in space.

Provided the unit vectors i,j,k parallel to the x,y,z-axis respectively, we can author a 3D vector valued function as:

r(t) = x(t)i + y(t)j + z(t)k

It can also be written as:

r(t) = <x(t), y(t), z(t)>

Both the above notations are equivalent and usually leveraged in several textbooks.Here

Space Curves and Parametric Equations

We defined a vector function r(t) in the preceding section. For differing values of t we obtain the corresponding (x,y,z) coordinates, defined by the functions x(t), y(t) and z(t). The group of generated points (x,y,z) thus, define a curve referred to as the space curve C. The equations for x(t), y(t) and z(t) are also referred to as the parametric equations of the curve C.

Instances of vector functions

This section displays some instances of vector valued functions that define space curves. All the instances are also plotted in the figure displayed after the instances:

• A Circle

Let’s begin with a simple instance of a vector function in 2D space:

r_1(t) = cos(t)i + sin(t)j

Here, the parametric equations are:

x(t) = cos(t)

y(t) = sin(t)

The space curve defined by the parametric equations is a circle in 2D space as displayed in the figure. If we vary from -𝜋 to 𝜋, we’ll produce all the points that lie on the circle.

• A Helix

We can go about extending the r_1(t) function of instance 1.1, to simply produce a helix within 3D space. We are just required to add the value along the z axis that alters with t. Therefore, we have the following function:

r_2(t) = cos(t)i + sin(t)j + tk

• A Twisted Cubic

We can also define a curve referred to as the twisted cubic with a fascinating shape as:

r_3(t) = ti + t^2j + t^3k

Derivatives of vector functions

It is simple to go about extending the idea of the derivative of a scalar function to the derivative of a vector function. As the range of a vector function is a group of vectors, its derivative is also a vector.

If

r(t) = x(t)i + y(t)j + z(t)k

then the derivative of r(t) is provided by r’(t) computed as:

r’(t) = x’(t)i + y’(t)i + z’(t)k

Instances of derivatives of vector functions

Its derivative is thus computed by computing the corresponding derivatives of x(t) and y(t) as displayed below:

x'(t) = -sin(t)

y'(t) = cos(t)

This provides us:

r_1′(t) = x'(t)i + y'(t)j

r_1′(t) = -sin(t)i + cos(t)j

The space curve defined by the parametric equations is a circle in 2D space as displayed in the figure. If we vary from -𝜋 to 𝜋, we’ll produce all the points that lie on the circle.

• A Helix

Just like the prior instance, we can compute the derivative of r_2(t) as:

r_2(t) = cos(t)i + sin(t)j +tk

r_2’(t) = -sin(t)i + cos(t)j + k

• A Twisted Cubic

The derivative of r_3(t) is provided by:

r_3(t) = ti + t^2j + t^3k

r_3’(t) = i + 2tj+ 3t^2k

All of the above instances are displayed in the image, where the derivatives are plotted in red. Observe the circle’s derivative also defines a circle in space.

More complicated instances

After you obtain a fundamental understanding of these functions, you can possess a lot of fun defining several shapes are curves within space. Other popular instances leveraged by the mathematical community are defined below and demonstrated in the figure:

The toroidal spiral

r_4(t) = (4 + sin(20t))cos(t)i + (4 + sin(20t))sin(t)j + cos(20t)k

The trefoil knot

r_5(t) = (2 + cos(1.5t)cos (t)i + (2 + cos(1.5t))sin(t)j + sin (1.5t)k

The cardioid

r_6(t) = cos(t)(1-cos(t))i + sin(t)(1-cos(t))j

Criticality of vector valued functions within machine learning

Vector valued functions have a critical part to play in machine learning algorithms. Being an extension of scalar valued functions, you would face them in activities like multi-class classification and multi-label issues. Kernel methods, a critical sphere of machine learning, can consist of computing vector valued functions, which can be later leveraged in multi-task learning or transfer learning.

Extensions

This section details some concepts for extension of the tutorial that you might desire to explore:

• Integrating vector functions
• Projectile motion
• Arc length in space
• Kernel methods for vector output