Coronvirus is spreading all over the world now. It’s a good time to go through the basic mathmatics by Shankar. There are 10 sessions in his book so I plan to document my learning progress in 10 blogs too. This is the first one – Differential Calculus.
- The first principle Chain rule derivative
problem 1.2.2 D(1/x) = -1/x^2 is deduced from the first principle chain rule, and then below
Another point is that to look at the following equation in a mathematical way.
Special cases such as f(x) = x^n,
The O part is the terms of order, divided both sides by delta x, the result is
In summary,
Definition of the derivative, derivative of a product of functions
D(fg) = gDf + fDg
a quotient of two functions
D(f/g) = [gDf – fDg]/g^2
chain rule for a funciton of a function
df(u(x))/dx = df(u)/d * du(x)/dx
The notion of the Taylor series is important. In his lecture, he gave an example f(x) = 1/(1-x) to illustrate how Tayler Series can approximate the values.
So the output is that 1/(1-x) = 1+x+x^2+x^3+x^4+x^5 …
Similarly, applying Taylors series on f(x) = (1+x)^n, one gets `1 + nx + n(n-1)*x^2/2!, the third term usually is chopped off, so it is 1 + nx.
The famous application is Einstein’s special relativity:
definition of hyperbolic function
Nature constant e (previous blog on how it is derived and its application in calculating interest profit in banks)
Applying Taylor series on e to the x,
Then the import cos and sin function
There is no restriction on x Vx, and one has to keep all the terms wihtout chopping off to make the approximation sensible.