Leibniz notation centers around the concept of a differential element. The differential element of is represented by . You might think of as being an infinitesimal change in . It is important to note that is an operator, not a variable. So, when you see , you can’t automatically write as a replacement .
We are dividing two numbers infinitely close to 0, and arriving at a finite answer. is another operator that can be thought of just a change in . When we take the limit of as approaches 0, we get an infinitesimal change .
Leibniz notation shows a wonderful use in the following example:
The two s can be cancelled out to arrive at the original derivative. This is the Leibniz notation for the Chain Rule.
Leibniz notation shows up in the most common way of representing an integral,
The is in fact a differential element. Let’s start with a derivative that we know (since is an antiderivative of ).
We can think of as the differential element of area. Since , the element of area is a rectangle, with as its dimensions. Integration is the sum of all these infinitely thin elements of area along a certain interval. The result: a finite number.
(a diagram is deserved here)
One clear advantage of this notation is seen when finding the length of a curve. The formula is often seen as the following:
The length is the sum of all the elements, , of length. If we have a function , the length element is usually written as . If we modify this a bit, we get . Graphically, we could say that the length element is the hypotenuse of a right triangle with one leg being the element, and the other leg being the element.
(another diagram would be nice!)
There are a few caveats, such as if you want to take the value of a derivative. Compare to the prime notation.
A second derivative is represented as follows:
The other derivatives follow as can be expected: , etc. You might think this is a little sneaky, but it is the notation. Properly using these terms can be interesting. For example, what is ? We could turn it into or . Either way, we get .
|Date of creation||2013-03-22 12:30:47|
|Last modified on||2013-03-22 12:30:47|
|Last modified by||mathcam (2727)|