Day 1
First of all, I'm not a math student. I have no associates with engineering yet. The reason why I started to read this book is simple, I wonder what calculus is. So we will see if it's possible to learn and understand calculus (at least start learning) with this book. It's highly recommended among engineering students. So I want to give it a chance. There are even examples, questions and practices in this book. So starting from totally 0, we will find out how far I'll progress. Who knows, maybe I will be so hyped to learn more and become a math student one day.
"What one fool can do, another can" -Ancient Simian proverb
First of all, before attempting to understand calculus, we should understand the two most important dreadful symbols. First one is d which means "a little bit of". Thus dx means "a little bit of x". Ordinary mathematicians like to say "an element of" instead of "a little bit of".
The second most important symbol is ∫ which is merely a long S, and may be called "the sum of".
So ∫dx means "the sum of all the little bits of x". The word "integral" simply means "the whole". If you think of duration of time for one hour, you may think of it as cut up into 3600 little bits called seconds. The whole of the 3600 little bits added up together makes one hour.
When we see these symbols it means we are going to perform the operation of totalling up all the little bits that are indicated by the symbols that follow. That's all.
We shall find that in our processes of calculation we have to deal with small quantities o various degrees of smallness. For example, there are 60 miutes in the hour, 24 hours in the day, 7 days in the week. There are therefore 1440 minutes in the day and 10080 minutes in the week. Obviously 1 minute is a small quantity of time compared with an hour. One minute is "one sixtieth" of an hour, and if you want to deal with even smaller quantities of time you should divide it 60 smaller parts, which, in Queen Elizabeth's days called "second minutes" Nowadays we call these small quantities of the second order of smallness "seconds". Quite an interesting fact, ain't it?
1/60 is a small fraction , then 1/60 of 1/60 (small fraction of a small fraction) may be regarded as small quantity of second order of smallness.
*The mathematicians ofthen use second order of "magnitude" (i.e greatness), when they really mean smallness. It can be confusing for beginners.
But sometimes small quantities can become important if multiplied with large factors. So they are not always negligible. Now in calculus we call all these dx du etc. as differentials. Differential of x and u. Such quantities like x.dx , x^2dx are not negligible. But dx.dx would be negligible, being a small quantity of the second order.