Calculus Like no other, Episode 1
If the above picture full of symbols and numbers makes you feel a little bit nervous and want to walk away from here, just continue reading. This is for you. In this series, I intend to explain and explore the terrors of differential calculus and integral calculus.
The first thing that comes to mind when people hear the word calculus is school tedious problems on calculus. It’s the fault of those teachers that don’t reside in the beauty underlying such methods.
I will take upon my shoulder to explain these beautiful methods in very simple terms and explore the connections and intuitions in some of the famous algorithms in the artificial intelligence and machine learning domain that use these beautiful methods.
What is the meaning of the two principal symbols that are used in Calculus?. d which merely means “a little bit of”. Thus dx means a little bit of x, or du means a little bit of u
∫ which merely is a long S, and may be called summation or the sum of. This means the sum of all the little bits of x. If you add all the little bits of x, i. e. dxs together, you get x itself. From a mathematician’s point of view, this is called the integral of.
So when you see an expression that begins with this terrifying symbol ∫, that gives you instructions that you are now to perform the operation of totaling up all the little bits that are indicated by the symbols that follow.
Degrees of Smallness
Throughout Calculs, we will have to deal with small quantities of various degrees of smallness. Let’s discuss a small exmaple to make things really clear.
There are 60 minutes 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 very small quantity of time compared with a whole week.
We needed to make more subdivisions of time, so we divided each minute into 60 still smaller parts, which, in Queen Elizabeth’s days, was called “second minutes” (i.e.small quantities of the second order of minute-ness).
Now if we fix upon any numerical fraction as constituting the proportion which for any purpose we call relatively small, we can easily state other fractions of a higher degree of smallness.
Thus if, for the purpose of time, 1/60 be called a small fraction, then 1/60 of 1/60 (being a small fraction of a small fraction) may be regarded as a small quantity of the second order of smallness.
Or, if for any purpose we were to take 1 per cent. (i.e. 1/100) as a small fraction, then 1 per cent. of 1 per cent. (i.e. 1/10,000) would be a small fraction of the second order of smallness.
hen we see that the smaller a small quantity itself is, the more negligible does the corresponding small quantity of the second order become. Hence we know that in all cases we are justified in neglecting the small quantities of the second—or third(or higher)—orders, if only we take the small quantity of the first order small enough in itself.
Now let’s apply these concepts to calculus. Now in calculus, we write dx for a little bit of x. These things such as dx, and du, and dy, are called “differentials,” the differential of x, or of u, or of y, as the case may be. If dx be a small bit of x, and relatively small of itself, it does not follow that such quantities as x·dx, or x2 dx, or ax dx are negligible. But dx×dx would be negligible, being a small quantity of the second order.
In Fig. 1, let us think of x as a quantity that can grow by a small amount so as to become x+dx, where dx is the small increment added by growth. The square of this is x2 + 2x·dx+ (dx)2. The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x2.
Thus if we took dx to mean numerically, say, 1/60 of x, then the second term would be 2/60 of x2, whereas the third term would be 1/3600 of x2. This last term is clearly less important than the second. But if we go further and take dx to mean only 1/1000 of x, then the second term will be 2/1000 of x2, while the third term will be only 1/1,000,000 of x2.
Now suppose the square to grow by having a bit dx added to its size each way. The enlarged square is made up of the original square x2, the two rectangles at the top and on the right, each of which is of area x·dx(or together 2x·dx), and the little square at the top right-hand corner which is (dx)2.
In Fig. 2 we have taken dx as quite a big fraction of x—about 1/5. But suppose we had taken it only 1/100—about the thickness of an inked line drawn with a fine pen. Then the little corner square will have an area of only 1/10,000 of x2, and be practically invisible. Clearly (dx)2 is negligible if only we consider the increment dx to be itself small enough.
Finally, An ox might worry about a flea of ordinary size—a small creature of the first order of smallness. But he would probably not trouble himself about a flea’s flea; being of the second order of smallness, it would be negligible.