Leibniz's notation

In calculus, Leibniz's notation, named after the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to show very small changes in x and y. These tiny changes are called infinitesimals. This notation helps us understand how things change and move, which is very important in math.

When we have a function where y depends on x, like y = f(x), Leibniz's notation helps us find the derivative. The derivative tells us how fast y is changing at any point. Leibniz showed that this change can be written as dy/dx, which looks like dividing a tiny change in y by a tiny change in x.

Even though the idea of infinitesimals was later replaced by more exact math rules, Leibniz's notation is still very useful. It works well in many situations and has been used in many areas of math and science. Today, this notation is a key part of how we study and solve problems in calculus.

History

The Newton–Leibniz way of infinitesimal calculus started in the 1600s. Newton used a method called fluxions, and Leibniz used sums and differences. Leibniz picked the integral symbol ∫ from an old style of writing the word "sum." He used the letter d to show changes, finally using dy for very small changes.

Leibniz first showed his integral sign in a paper in 1686 and used dx in another paper in 1684. Even though some later mathematicians found new ways to understand these ideas without very small changes, Leibniz's notation is still used today. It is helpful, especially when solving some kinds of math problems. It also works well when checking the sizes of different parts in physics problems.

Leibniz's notation for differentiation

Main article: Notation for differentiation

Suppose a number y shows how another number x changes. We can write this as y = f(x). In Leibniz's way of writing, the rate at which y changes compared to x is shown using special symbols: dy/dx.

There are other ways to write this rate, like using a small mark called a prime (y') or a dot over the letter for time changes (like x˙ for how x changes over time). Leibniz's way has stayed popular because it helps us understand many math ideas clearly.

When we want to find how the rate itself changes, we can write this using more of Leibniz's symbols, like d²y/dx² for the rate of the rate. This helps us study even more detailed changes in how numbers relate to each other.

Leibniz's notation for integration

Leibniz introduced the integral symbol that we use today for integration. He first wrote about this symbol in his private notes in 1675. He shared it in a paper called "De Geometria Recondita et analysi indivisibilium atque infinitorum" in Acta Eruditorum in June 1686. He chose the symbol because it looked like an infinite sum of very small parts.

Use in various formulas

Leibniz's notation helps remember important math formulas. For example, the chain rule can be written as:

dy/dx = (dy/du) × (du/dx)

This works with many connected functions. Another example is integration by substitution, where a new variable helps solve the problem.

When a function can be reversed, its derivative can be found using:

dx/dy = 1 / (dy/dx)

These notations often look like simple fractions, which makes solving calculus problems easier.

Modern justification of infinitesimals

In the 1960s, mathematicians found new ways to explain the very small numbers Leibniz used. These new ways help them follow modern rules. Some mathematicians use these ideas today. One book even teaches calculus with these tiny numbers.

Today, tiny changes in numbers help us understand how things change and add up. Most mathematicians think about these tiny changes in different ways. To use these ideas, we need to think of numbers in a bigger, more detailed way.

Other notations of Leibniz

Leibniz tried many different symbols in math and thought good symbols were very important. In a letter in 1693, he said that using the right symbols was a key part of solving math problems.

Over time, Leibniz improved his ideas about symbols. He wanted symbols that could fit neatly on a page. At first, he used a special line called a vinculum to group symbols together, but later he used parentheses instead. This made printing easier and looked nicer.

Leibniz created more than 200 new symbols that we still use today. Besides the small changes in x and y (dx and dy) and the integral sign ( ∫ ), he also introduced symbols like the colon (:) for division, a dot (⋅) for multiplication, and signs for similarity (~) and congruence (≅). He used Recorde’s equal sign (=) instead of an older symbol for proportions, and he developed a special notation for determinants.