Do you know the position where God proclaimed that let there be Universe?

Hi Guys

A particle is a point mass. To study its motion we must know its position with respect to a desired point  in one dimension (1D) along a straight line , or two intersecting lines ( axes ) forming plane in 2D or three intersecting lines (axes) forming space in 3D.

Rene desCartes (1596-1650) invented the method of studying Geometry using Algebra by defining Co-ordinate System.

In 2D plane ( an ant be crawling on a floor ) you need two intersecting lines to determine the position of the ant uniquely, called axes. The point of intersection is called origin and the distances of the point from the axes gives us an ordered pair (a,b) called the co-ordinates. desCartes made it easier by considering the axes to be orthogonal ( mutually perpendicular ) and there evolved the analytical geometry with rectangular cartesian system. You know graphs, so you have some idea about it.

Here the ordered pair (x,y) are the coordinates where x is the distance from Y-axis called abscissa or X-coordinate and y is the distance from X-axis called ordinate or Y-coordinate. Thus the coordinates of origin is (0,0).

Identically for 3D space ( a mosquito by flying in that room ) we need another coordinate - distance of the point from floor (XY-plane) called Z-coordinate

and we get a point in space as an ordered triad (x,y,z).

You may understand easily the difference between 3D and 2D as the flying mosquito in the room and the shadow of the mosquito on the floor.

Now I give you something to think on :

Can a particle be at two or more positions at an instant of time?

or,  if a particle changes its position will there be a finite time elapsed during its motion?

So now you see that to study motion we must know the Coordinate Geometry well.

All the best. You are ready now to get into the realms of Physics...

Purpose to learn is always important

Hi Guys

I believe to learn a subject one must know the purpose - why should we study that particular subject?

Let me ask you then why should we study Trigonometry?

This is for our basic instinct...to grab as much land on earth as possible, since ancient times to this date people made their index of power as procurement of larger land. Thus a King gave the object of measurement of his land ( whether that is greater than his rival neighbouring states ) to his department of wisemen; there were two objectives to know - the area ( for his satisfaction/grievance ) and the perimeter ( for defence arrangement ).

In Greece philosopher mathematician Euclid at around 300 BC started his journey to organise the subject of Geometry 
He is called the
Father of Geometry

In fact to me this was the start of 
human beings in deductive
logic. He studied the
most important
of plane figures
- the triangle or
trigon (polygon
with three sides.
He noted that
a triangle
has  some
features:
1. It is a plane figure bounded by least (3) number of straight lines
2. Any polygon can be divided into triangles; any irregular area can be considered to be broken into triangles subject to a very close approximation. So if we know triangles in detail - its area and perimeter, then we know any surface and its boundary (the King's requirement).
3. A triangle has three sides and three angles
4. The sum of all its three angles is two right angles
5. Any two sides are greater than the third one
and finally
6. greater is the side when greater is its opposite angle

The last property was interesting for the wisemen then to find a relation between the sides and angles.

........is it in direct proportionality?  No, there was no match found but already there was that extraordinary Theorem of Pythagorus for a special triangle - right angled triangle in which we have a relation between three sides. Pythagorus ( 570 BC ) was also
a polymath.

Here started the subject of Trigonometry with an intention to find the relation between the sides and angles of any general triangle - scalene. This is somewhat an induction process in forming definitions of trigonometric ratios - sine[sin], cosine[cos], tangent[tan] - in between sides of a right-angled triangles. With the growth of the subject we obtain the Sine Rule in general for scalene.

Sine rule says that a side is proportional to the sine of its opposite angle. We also find area [ Hero's Formula ] and perimeter through a chapter called Properties/Solutions of Triangles in Trigonometry.

Later we see that the measurement of the heights and distances of mountain peaks or other objects were done with the help of Trigonometry. In 1831 Bengali mathematician, Radhanath Sikdar used Trigonometry to find the height of Peak XV as the tallest mountain of the world under a leadership of the surveyor, George Everest. Now you know the name of the peak in the Himalayas.

However, solving the purpose Trigonometry did not get stalled over there; later we find these ratios as trigonometric functions which contributed heavily for the advancement of Mathematics.

Next day I shall speak on the purpose of  learning Co-ordinate Geometry.

All the best

God gave the digits, man made the rest...

My dear friends

I could not post yesterday. Actually last portion on Maths - Calculus is still pending to be discussed. Calculus is developed independently and at the same time by both Newton[1643-1727]  ( at your left)             and     
Leibnitz [1646-1716]  ( down and at your right ) at the end of seventeenth century. Newton applied the same to explain Motion and Gravitation.

Calculus is a latin word meaning stones for counting. Calculus is the branch of Mathematics that deals with the study of changes.
Let us consider that a car be moving along a straight road with constant speed.  If we want to know the distance covered at a given interval of time then we just calculate the product of that speed and time interval. But in general if the car moves with variable speed and along any path, the technique described above cannot solve the problem.

First we must know the speed of that car at each point of its path; we call it instantaneous speed. To determine this we break the path into very small fragments of distances that are travelled in even smaller intervals of time. So speed, in any such small interval, is the ratio of that description of small distance to the very small time interval. This is the speed during that time interval but still not instantaneous. We know that there is no end to reach smallness i.e., whatever small quantity you may think, I can  think even smaller. That is why we call it infinitesimally small. As a matter of fact we approach towards zero but not exactly zero. 

Now if we approach zero for that time interval, we get that ratio as the instantaneous speed. This ratio, subject to the condition of approaching zero, is called the derivative and determination of this derivative is the primary objective of Calculus through its one branch - Differential Calculus. The process to find the derivative is called differentiation. Basically derivative in general is the rate of change of a function (distance) per unit change of the independent variable (time) on which the function is dependent, at a desired value of the variable (time). This is why derivative is also called the rate measurer and Calculus is called study of changes. In every sphere of life we have a dependence to study and is therefore, the importance of the subject.

However, it is not finished yet. We still  need to know the total path described by the car with variable speed during a given finite time interval. For that we need to sum up each product of that ratio  [small distance to the infinitesimally small time interval] and that infinitesimally small time interval all over the path. The summation is called integration and this other branch is called Integral Calculus.

In differentiation you measure the function at each point by breaking whereas in integration you sum up to know the function along a finite interval. Integration is thus reverse process of differentiation.

I hope you now have got some idea about the purpose of this subject. I have seen in many cases the students learn Calculus mechanically without knowing its necessity and purpose.

Please do not learn this branch of Mathematics just for your examination. Be positive. If you love Mathematical Analysis, I am sure that, you will collect enough fuel to go for without looking back.

Wish you all the best

  

Time is the essence...

Hi Guys

I am sure that you now have found me a great bore! Monotonous directives, cliche treatments and rigorous Maths are making your life hell again. Let us discuss something different today. Let us remember a great philosopher ( in ancient times philosopher means a wise person who has authority in his subject ).

You know Human Civilization is nearly 4000 years old at the maximum, whereas the age of Earth is nearly 4500 million years. So our civilization is a tiniest of ticks in the Big Clock. Still there is someone, an extra-ordinary human being in wheelchair, who claimed that if we know the exact status of that precambrian soup, immediately after God has fired His oven to cook  this Universe, then someday we( this human race ) shall also be able to create another! God is God, but His son is no mean.

Rabindranath while dedicating his book on science "Bishwaporichoy" to Satyendranath Bose wrote that

 'human beings are the only living creatures who always questioned his own understandings, refuted illogical reasonings and always found to be happy in defeating contradictions...they always entered into a world beyond their limits to perceive but  came up with the truth'

To start with such human beings I remember the greatest of polymaths, Leonardo da Vinci (1452-1519). Have a look at his self portrait, now it is placed at the Royal Library of Turin. It is drawn in red chalk and most probably in 1512, a few years before his death. 

The portrait depicts a look of the genius that he is not happy with all he has done. May be some more extra-ordinary inventions he thought of but could not finish, may be he is not happy for what he has done...still he is amongst  those few sons of God. They are hand picked by the Father. I am too ignorant to describe even the complete works of  Leonardo. He is the archetype of the 'Renaissance Man' who was a painter, sculptor, architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, botanist and writer.

Phew...still he is not satisfied. That is the sign of a genius.

And also think of the span of time...all he had done in 67 years. We do not have much time left. If you really want to make a mark please do not waste your time.

Ramkrishnadev said his followers to make a mark. That is the meaning of your existence.

This is not easy now. In last 500 years the height of the pillar of civilization is no small because there are too many contributions from so many geniuses. You have to reach to that top of your branch and then make that mark.  Reference level is high and time is short.

So can you waste your time, my friends?