The Two Symbols of Calculus

Mathematics / Calculus

The Two Symbols of Calculus

The preliminary terror which chokes off most people from even attempting to learn calculus can be abolished once and for all by simply stating what is the meaning — in common-sense terms — of the two principal symbols that are used.[1]

These dreadful symbols are:

Symbol 1: $d$ — "A Little Bit Of"

The letter $d$ merely means "a little bit of."

Thus $dx$ means a little bit of $x$. And $du$ means a little bit of $u$. Mathematicians prefer to say "an element of" instead of "a little bit of." Just as you please. But you will find that these little bits (or elements) may be considered to be indefinitely small.

: "A little bit of" — the differential operator: Whatever quantity we're taking a little bit of

You read $dx$ as "dee-eks", $du$ as "dee-you", $dy$ as "dee-wy."

These are not two separate things multiplied
$dx$ is a single symbol — it does not mean $d \times x$. It means "an infinitesimally small piece of $x$." The $d$ and the $x$ are inseparable. Think of it like the word "a**n**" — the letters work together, not apart.

Symbol 2: $\int$ — "The Sum Of"

The symbol $\int$ is merely a long S, and may be called "the sum of."

Thus $\int dx$ means the sum of all the little bits of $x$. And $\int dt$ means the sum of all the little bits of $t$. Mathematicians call this symbol "the integral of." The word "integral" simply means "the whole."

: "The sum of" — add up all the little bits: Each infinitesimally small piece of $x$: The whole of $x$, reconstituted from its pieces

Now any fool can see that if $x$ is made up of a lot of little bits, each of which is called $dx$, and you add them all up together, you get the sum of all the $dx$'s — which is the same thing as the whole of $x$.

Time in Seconds
If you think of the duration of 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 make one hour. $$\int ds = 3600 \text{ seconds} = 1 \text{ hour}$$ The $\int$ just means: "add them all back up."

That's All

When you see an expression that begins with the terrifying symbol $\int$, you will henceforth know that it is put there merely to give you instructions that you are now to perform the operation of totalling up all the little bits that are indicated by the symbols that follow.

And when you see $dx$ or $dy$ or $dt$, you know it just means a little bit of that quantity — indefinitely small, but not zero.

That's all.[1]

Sources

[1]Thompson, S.P. — Calculus Made Easy, 2nd ed. (1914), Ch. Ihttps://www.gutenberg.org/ebooks/33283