Notation & Vectors

Mathematical notation & an introduction to vectors.

Mathematical Notation

Here we'll talk about standard notation used in math and physics. A lot of Greek letters are used, so it's like you're learning another language. In fact, once you've learned math, you've learned the language of physics!

When you see a bolded number in parentheses (for example (1)), it is referring to the equation on that line. It makes for easier referencing to simply say "See equation (1)" than to write out the whole equation.

Delta: A change in quantity

The Greek symbol delta, D, is used to indicate a change in quantity. For example, if an object moves from position to we would write its change in position as

Derivatives

A derivative is a quantity that is obtained from a different quantity. For example, velocity is a derivative of position—if you know how an object's position has changed over time, you can derive its velocity. Similarly, if you know how the object's velocity has changed over time, you can derive its acceleration.

Let's consider average velocity. It is a change in position divided by a change in time:

or . (1)

Instantaneous velocity, v is the limit of (1) as or as .

Written with the notation of a derivative, velocity becomes

. (2)

We read it as "velocity is the derivative of x with respect to t."

In most cases, x will be some function of t, written as and read as "x as a function of t" or "x of t."

Now let's take a look at acceleration. Similar to our discussion above, average acceleration is a change in velocity divided by a change in time:

Instaneous acceleration, a, is the limit as .

Written as a derivative, we have . (3)

Similarly to position, velocity is a function of t, or .

Notice what happens when we substitute equation (2) into equation (3):

(4)

Thus, we see that acceleration is a second derivative of position.

In summary:

1. Velocity is a derivative of position with respect to time: (2)

2. Acceleration is a derivative of velocity with respect to time: (3)

3. Acceleration is a second derivative of position with respect to time: (4)

Vectors

Vectors represent quantities that have two characteristics: magnitude and direction. Position, velocity, and acceleration are all vector quantities. In the above discussion, I was a little loose with my language for illustrative purposes. I'll clarify things here.

Position refers to an object's location on a coordinate system. In a 3D system, there are three points to that location: an x-coordinate, a y-coordinate, and a z-coordinate, denoted as (x, y, z). Position is a vector because it describes how far from the origin the object is (magnitude) and also in what direction.

In the above illustration, the black point is at coordinate (x,y,z) at a distance d from the origin. Its position is thus represented by a magnitude (d) and a direction ( (x, y, z) ).

Velocity is vector because it is a change in position, which is also a vector. Its magnitude is the speed of the object, and its direction is the direction of the motion.

Similarly, acceleration is a vector because it is a change in a vector quantity—velocity.

The notation for vectors is either boldface (v) or with an arrow over the top of the letter ( ).

In summary:

1. Vectors must include a magnitude and a direction.

2. Vectors are denoted as either boldface or with an arrow.

3. Position, velocity, and acceleration are vectors.

Scalars

Another term with which we'll want to be familiar is scalar. A scalar quantity has magnitude, but no direction. I've mentioned a couple of scalar quantities already: distance and speed. Distance is simply a measure of how far from an origin point and object it is; there is no indication of direction. Similarly, speed is how fast an object is moving; direction is not indicated.

For example, to state the velocity of your vehicle you would say "70 miles-per-hour, westbound." There is both magnitude (70 mph, or speed) and direction (westbound.)

If you say you are traveling at 70 mph, you have given a scalar quantity (speed), not a vector (velocity) because you have not specified a direction.

If you tell your friend that you are 30 miles from their house, you have given a scalar quantity (distance). If your friend wants to know your position (a vector), you would have to include a direction such as "30 miles north of your house."

In summary:

1. Scalars are numerical quantities that are a magnitude only.

2. Distance is the magnitude of position; it is a scalar.

3. Speed is the magnitude of velocity; it is a scalar.

Conclusion

In everyday conversation we tend use speed and velocity interchangeably; or distance and position interchangeably. However, there are significant differences when you use these terms in a math or physics setting.

When in doubt, just remember supervillain Vector, from Despicable Me:

He's "committing crimes with both magnitude and direction!"

For The Interested Reader

The ratio of a circle's circumference to its diameter has been approximated to 3.1416. An ancient Babylonian tablet dated around 1900-1680 BC records this ratio as 3.125. The ancient Egyptians give an approximation of 3.1605 in 1650 BC.

In the 1700s mathematicians began using p to represent this ratio. Leonard Euler popularized its usage in 1737. It has since been defined as this ratio, a constant that is irrational and transcendental.

p even has its own holiday on March 14 (3/14)—known as Pi Day.

For The Interested Reader

The concept of vectors did not arise until the 19th century. Mathematicians conceived of complex numbers (that is, numbers with both a real and imaginary component) as points in 2D space. The result was a vector representation, but the notation wasn't widely used.

James Clerk Maxwell, father of the theory of electromagnetism, divided physical quantities into scalars and vectors in the 1870s.

In 1881 J. Willard Gibbs, a Yale professor, started teaching vector analysis and vector algebra as we know it today.

For The Interested Reader

This is just an introduction to derivative notation. For a more complete discussion on the use of derivatives in mathematics, check out this lesson.