## Thursday, 8 November 2018

### Graphs and Level curves of functions of Two variables

Definitions
• The set of points (x,y) in the plane where a function f (x,y) has a constant value f(x,y)=c is called a level curve of f
• The set of all points (x,y,f (x,y)) in space, for (x,y) in the domain of f, is called the graph of f
• The graph of f is called the surface z=f (x,y)
• The equation of the following hyperbolic paraboloid is z=50+x^2-y^2

## Tuesday, 30 October 2018

### Graphs and Level curves of functions of Two variables

Definitions
• The set of points (x,y) in the plane where a function f (x,y) has a constant value f(x,y)=c is called a level curve of f
• The set of all points (x,y,f (x,y)) in space, for (x,y) in the domain of f, is called the graph of f
• The graph of f is called the surface z=f (x,y)
• The equation of the following paraboloid is z=50-x^2-y^2
Extracted surface of Paboloid using Matplot Python

 Mesh Grid -Extracted Surface of Paboloid

## Wednesday, 24 October 2018

### Inverse Trigonometry

\color{brown}"Proof : " cos(2cos^{-1}x)=2x^2-1
put x= cos u
ie cos^{-1}x=u
then LHS=cos(2 u)=2cos^2u-1=2x^2-1

## Sunday, 14 October 2018

### Topic -Refractive Index

Sai's classes now giving you notes on selected topics  (problems with solutions)  Intending to support Secondary school students The following topic is on Refraction of light.Click the link below to get notes in pdf format and you can download it.Update will be done at intervals

## Sunday, 7 October 2018

### x to the power i -How to handle Complex power ?

\color{brown}"The following steps explains i in power"

x^{a+ib}=x^{a}.x^{ib}=x^{a}.(e^{log(x)})^{ib}because x=e^{log(x)}

=x^a.e^{ib.log(x)}=x^{a}[cos(b.log(x))+isin(b.log(x))]  (by de Moivres theorem)

In the above result put a=0 and b=1 results in

x^i=[cos(log(x))+isin(log(x))]

## Wednesday, 3 October 2018

### Gradient map of Fourier Series - Square Wave

The Fourier series used to create a square wave function Only harmonics 1ω,3ω,5ω  are used for the plot .By increasing the number of harmonics the wave become more like a square wave
f(t)=\frac{4}{\pi}[sin(2\pi f t)+\frac{1}{3}sin(6\pi f t)+\frac{1}{5}sin(10\pi f t)]
 fig1:Square wave function from Fourier series
The figure below (fig:2) the red spectral line indicates the crest and blue  shows the trough of the square wave And inside the red and blue line the fine lines shows the gradient map ; the harmonics So it is an excellent way to represent the superposition of waves
 fig2: Matplotlib pcolor for the same Fourier transform as in fig1