PHY-101: Introduction to Vector Calculus

 Beginning of a New series

Welcome back to my blog. Over the months I have posted many blogs on different things in space be it the infamous space race between the US and USSR, the difference between an asteroid or meteor, or be it what are stars and what they are made of etc. But I think it is time for us to dive a bit deeper and understand how it all came through together. Why do we study physics? Why ask questions about the observable universe, and in extension why ask these questions at all?

I will take you from the start of physics which is through mathematics to the world of vector calculus, from then on we will discuss Classical mechanics, Electricity and Magnetism extending to Electromagnetic Theory, Thermodynamics, Solid State Physics, Quantum mechanics and a bit of Astrophysics. (Not necessarily in that order) So hold tight for we will first start with arguably the most important and boring part of the series(for some do not like the math).

Let us begin with Vector Calculus!

1. Introduction of Vectors

What are vectors?

In physics, anything with a magnitude and direction is deemed a vector. The simple way to imagine this is pushing or pulling off the door, a task we normally do. When we push the door we apply some Force in a direction towards the door, thus the force is a vector. The way a vector is represented is \(\overrightarrow{A}\). This is read as vector A.

The standard way to write a vector is \(\vec{A}=a\hat{i}+b\hat{j}+c\hat{k}\), where a, b and c are the components or rectangular components of \(\vec{A}\) in the x, y, z direction respectively.

There are different types of vectors and we will not go into much detail for that would be time-consuming and something you can read in a book. (I'll drop in a book recommendation for you below) But what you need to know is about the unit vector.

A unit vector is basically a vector of magnitude 1 and some arbitrary direction often written as \(\hat{A}\).

and, $$\hat{A}=\frac{\vec{A}}{|A|}$$, $$|A|=\sqrt{A_{x}^2+A_{y}^2}$$ assuming that we are dealing a problem in 2D. (that is why we have used the x and y components)

Now, that we have an overview of what are vectors we might as well shift our attention towards scalar quantities.

A scalar is something that only has magnitude but no direction. The factor \(|A|\) in the division of the unit vector is the magnitude of the vector which is scalar.

Some of the quite obvious examples of each are as follows:

1. vectors: velocity, force, acceleration, torque, momentum, etc

2. scalar: speed, charge, temperature, mass, energy, etc.

There is another set of quantities called the tensors. They are neither Vector nor Scalar. Examples include Stress, Strain, Elasticity, etc. We will for the duration of this series not indulge ourselves in tensor quantity.

A vector can be divided into its components. Assuming we are in a rectangular coordinate system which has three axes, x, y, z. Like the magnitude shown in the above equation, we have used the subscript of x and y to denote the magnitude of the vector in that direction. Regardless of what coordinate system you use, we can separate the vector into its component form in each of the respective directions.

Then we also have scalar and vector fields which can de defined very intuitively, if for a point in region R say (x,y,z) we have a corresponding scalar \(\phi(x, y, z)\) then it is a scalar field like that of a temperature, but if that point corresponds to a vector say \(\vec{V}(x, y, z)\) then it is called a vector field. 

2. Scalar and Vector Product

Unlike simple mathematics where we just multiply two numbers together to get the answer to a multiplication problem in physics or vector calculus, we have two kinds of multiplication, scalar and vector.

Scalar Product

Often called a Dot product is defined as the product of the magnitude of vectors and the cosine between them. $$\vec{A}.\vec{B}=|A||B|cos\theta$$ This product of vectors gives a result that is a scalar quantity.

Vector Product

This is a cross product of vectors which is defined as the product of magnitude of the vectors and the sine between them in the direction mutually perpendicular to the vectors.$$\vec{A}×\vec{B}=|A||B|sin\theta\hat{n}$$

From the definitions above we can conclude on some important results.

1. When a vector is in dot product, the like components(unit vectors) multiply and equal 1 while unlike components equals 0.
2. If \(\vec{A}.\vec{B}=0\), and A and B are not null vectors then A is perpendicular to B.
3. When a vector is in the cross product, the like components multiply and equal 0 and unlike components give the perpendicular component, eg \(\hat{i}✕\hat{j}=\hat{k}\).
4. The magnitude of \(\vec{A}×\vec{B}\) is the same as the area of the parallelogram with sides A and B.
5. If \(\vec{A}×\vec{B}=0\), and A and B are not null vectors, then A and B are parallel.

We also have scalar and vector triple products. But that I leave as an exercise for you to find. (Have seen many authors do this before)

Conclusion:

I feel this is a good time to stop for the first blog and recapitulate what we have covered. So we started our time by knowing what are vectors, scalars and tensors. We then proceeded to understand what we mean by components of vectors and what vector and scalar fields are. Then we understood the difference between dot and cross product where we have two very important cases in the results i.e. 2 and 5. Both will be used many times in derivations directly.

Next time we will start with Differentiation and Integration, two very strong weapons for physicists while tackling a problem. Then understand different types of curvilinear coordinate systems.

Furthermore, we will begin our journey to Classical Mechanics!

Book Recommendation:

Vector Analysis Schaum's Outline by Murray R. Spiegel