Calculus -Newton and Leibniz techniques
In this post i want to discuss about how calculus was perceived by Newton (derivatives) and Leibniz (differentials) and difference in their techniques
All Images used in article has been Taken from Book Infinite Powers by Steven Strogatz
First I would describe Fundamental Theorem of Calculus and then how Newton and Leibniz went about computations for same
Fundamental Theorem of Calculus
Two things fundamental theorem describes
- Relationship between curve and its rate of change ( slope of curve), forward problem
- Relationship between curve and area it accumulates , backward problem (Integration)
Further both of these can be looked up as follows
- Rate of change - How much a quantity changes , Change at any Instant , represented by Slope
- Area Under Curve - How much a quantity accumulates over period of time
Lets look at how two great scientists arrived at calculations using different techniques.
Leibniz’s Technique
Leibniz introduced 2 new concepts
- Differentials — small differences in x and y.
- Infinitesimals -Very small quantity, very small change.
- Rate of Change
2. Area Under Curve
Total Area under curve is sum of individual areas
which can be expressed using Integral symbol as below
Modern day calculations use above techniques. One good aspect of Leibniz’s technique is it has made the process mechanical.
However a drawback is infinitesimals dont exist . They are just abstract entities used for computation.
Now lets look at how Newton approached the same
- Rate of Change
Consider a small duration of , In this case x changes via dx, but since duration is small, we can assume y to remain constant. Thus area(dA) accumulated during this time
⇒ dA = y dx
⇒ dA/dx = y
Which is formula for rate of change of Area vs x.
2. Area Under Curve
Newton devised technique called Quadrature of Curves to compute area under curve
There were 2 Rules in this technique
Rule 1
Rule 2
If there are multiple terms in above equation, then area will be made of each of above terms
Further to this if curve is not of form below, it needs to be converted in below form
Below are steps to transform the equation
- Convert any equation to above form y = f (x)
- If equation needs to be expanded/converted to Power Series - Apply binomial Theorem .
- then Apply Rule 2 above
Examples of Quadrature of Curves
In modern day calculations usually technique specified by Leibniz is used because its very mechanical and straight forward
References
- Infinite Powers — Steven Strogatz
- Journey Through Genius — William Dunham
