Functions describe the relationships between entities, and are extremely useful to explain the causality of one quantity on another.
Now some quantities require a set of numbers to be represented accurately. These are known as vector quantities, and occur everywhere.
Combining these two highly important concepts, what we get are vector valued functions, which not so surprisingly are crucial for us to understand and influence the world we live in.
For instance, magnetism behaves according to vector valued functions, and so the concept is needed to study natural events like the northern lights. The understanding and taming of those functions have further given us magnetic resonance imaging, a revolutionary tool for medical diagnosing.
The orbit of a satellite around the earth is a vector-valued function, since the position of the satellite is described using a vector with 3 components
A vector-valued function can have different amount of input variables. For our satellite, the position only depends on a single variable .
The position in the plan at time can be written as
taking the derivative of each component yields the vector for velocity
taking the derivative yet again yields the vector for acceleration
Vector valued functions
A vector valued function
is a function which takes one variable and maps it onto two variables.
It's not as bad as it looks. Let's break it down with an example: imagine a car. Driving over a large, flat field on a winding road, its position can be described by a position vector .
The position and thus both variables depend on the time , so we get .
While pondering upon this monumental fact, we introduce some notation. Recall that the unit vectors in the , and directions respectively can be written as , and . Other fairly common notations are , and , or , and .
Thus all of these are the same, and represent the position as a function of the time variable :
The function doing the mapping is commonly described with a funky little arrow:
It just means that it takes as input and outputs the coordinates as a function of .
In 3D and beyond
The little car of ours leaves the field behind and crawls up a hill at the outskirts of a mountain chain. We require a third coordinate for the level above the sea, so we grab the next one: .
Like the and the , depends on the time. So, now we have a function
This generalizes to higher dimensions. Imagine the car traveling in the -dimensional space . Okay, we may have to drop the car at this point. Nevertheless, there are functions like this:
Here, for simplicity we drop , , and so on and call all the coordinates with the ordered indexes.
As you may recall, this type of function, mapping one variable onto several, is commonly used for parametrizing curves.
Derivatives of vector valued functions
Back into the car
In the last lecture note, when we left our car it was driving into the mountains. Not being a self-driving car, we'd better get back in to assure that it doesn't drive off a precipice.
So let's take it one step further. The position of the car was:
Say we'd like to know the velocity. We can read it off the speedometer, but that's no fun compared to what we'll do.
Taking the derivative
Have a look at this curve. It describes the motion of, for example, our car along the road-curve .
Observe how the position vector changes as is incremented by . Calling the small change-in-position vector , we see that over becomes:
Letting go to the infinitely small differential gives the derivative:
That's the velocity right there. Notice that as we make the time difference smaller, the difference vector will be more and more parallel to the velocity vector. The scaling done by dividing by makes the derivative and the velocity identical.
Now we know that can be separated into coordinates, all of which just depend on . Therefore:
Thus, the derivative of a vector valued function can be taken in one direction at a time. This generalizes to higher dimensions.
Sometimes, we need to combine vector valued functions with each others and/or with scalar valued functions. Just a reminder: a scalar valued function outputs a scalar.
Let and be differentiable vector valued functions and a differentiable scalar valued function. Then, the combinations , , and are all differentiable, and obey the following differentiation rules:
Note that these rules all follow our intuition from differentiation of scalar functions. , and are versions of the product rule, and is the chain rule. The last rule we get from applying the chain rule to .