AND NOW TO GRASP EULERS THEOREM, ONE MIGHT HAVE TO GO DOWN A SLOPE.....
But in the sense you might have to let your thinking go down the slope.
For Eulers Theorem is not so complicated you see....its simple.
EULERS THEOREM
COS (
COS (0) -- ----------------> A measure of how much horizontal an object is.
SIN (0) ---------------------> A measure of how much vertical an object is.
i------------------------------> inclination present in the object.
COS(20) + i * SIN (20)
MORE ON EULERS THEOREM.
|||| Eulers theorem is all about inclination or rotation (actually,....to incline an object, you rotate it first and stop. And to rotate an object,...you continously keep on inclining it without stopping . So,...both of them are the same actually.)
I never understood Eulers Theorem initially.
It was only after I understood the meaning of complex numbers after i came across Kalid Azad from betterexplained.com that Eulers theorem clicked.
By this time I had come to realize already that Sin( ) is a measure of how much perpendicular an object is and Cos( ) is a measure of in alignment or parallel an object is.
So if you are measuring the slope or the inclination of anything w.r.t the ground....for example how much is the slope of a staircase with respect to the ground....it will be
cos( ) + i.sin( ).
For example if the slope of the staircase is 30 degree....the horizontal component of it will be cos(60) = 1/2
And the vertical component of this staircase will be sin(60)= sqrt(3)/2.
Together it will be e^i(60) = cos(60) + i.sin(60)
e^i(60) = 1/2 + i.sin[sqrt(3)/2]
If the actual length of the staircase is 100 metres say....then according to Eulers theorem it will be.
100 x e^(i.60) = 100.cos(60) + i.100.sin(60)
100 x e^(i.60) = 100 x 1/2 + i.100 x sqrt(3)/2
= 50 + i.(86.66)
50 or 100cos(60) is actually the front-view(the length of the staircase as seen if you stand exactly in front of it) of the staircase.
86.66 or 100sin(60) is actually the bottom-view (the apparent length of the staircase as seen by the eyes if you stand just below it) of the staircase.
DOES EULERS THEOREM INDICATE ROTATION OR INCLINATION?
Well actually rotation and inclination are inseperable from each other. They are actually one and the same.
Can you rotate anything without inclining it? And likewise,can you incline anything without rotating it?
One may say that but the staircase was made inclined right from the time it was created. Did someone make it first flat and later incline or rotate it?
Well if you trace back to the birth of the staircase....it was made from say bricks...the worker who made it had to arrange the bricks in an inclined manner in order to create it the way it is.
Again rotation is inclining something continously.It is the same phenomenon , just seen from two different angles.
EULERS SPIRAL
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THE ABOVE IS A EXCERPT FROM A BOOK AVAILABLE FOR PURCHASE (5$) AT THE FOLLOWING LINK
I hardly understood Math in school. In fact was on the verge of dropping the subject I loved the most because as much as I loved the theory of it, I could not understand the math involved in it. However I loved the subject too much to be able to live without. Hopelessly, I was continuing my love-affair with it.
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CONTACT
binnoypanicker@gmail.com
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COS(20) + i * SIN (20)
MORE ON EULERS THEOREM.
|||| Eulers theorem is all about inclination or rotation (actually,....to incline an object, you rotate it first and stop. And to rotate an object,...you continously keep on inclining it without stopping . So,...both of them are the same actually.)
I never understood Eulers Theorem initially.
It was only after I understood the meaning of complex numbers after i came across Kalid Azad from betterexplained.com that Eulers theorem clicked.
By this time I had come to realize already that Sin( ) is a measure of how much perpendicular an object is and Cos( ) is a measure of in alignment or parallel an object is.
So if you are measuring the slope or the inclination of anything w.r.t the ground....for example how much is the slope of a staircase with respect to the ground....it will be
cos( ) + i.sin( ).
For example if the slope of the staircase is 30 degree....the horizontal component of it will be cos(60) = 1/2
And the vertical component of this staircase will be sin(60)= sqrt(3)/2.
Together it will be e^i(60) = cos(60) + i.sin(60)
e^i(60) = 1/2 + i.sin[sqrt(3)/2]
If the actual length of the staircase is 100 metres say....then according to Eulers theorem it will be.
100 x e^(i.60) = 100.cos(60) + i.100.sin(60)
100 x e^(i.60) = 100 x 1/2 + i.100 x sqrt(3)/2
= 50 + i.(86.66)
50 or 100cos(60) is actually the front-view(the length of the staircase as seen if you stand exactly in front of it) of the staircase.
86.66 or 100sin(60) is actually the bottom-view (the apparent length of the staircase as seen by the eyes if you stand just below it) of the staircase.
DOES EULERS THEOREM INDICATE ROTATION OR INCLINATION?
Well actually rotation and inclination are inseperable from each other. They are actually one and the same.
Can you rotate anything without inclining it? And likewise,can you incline anything without rotating it?
One may say that but the staircase was made inclined right from the time it was created. Did someone make it first flat and later incline or rotate it?
Well if you trace back to the birth of the staircase....it was made from say bricks...the worker who made it had to arrange the bricks in an inclined manner in order to create it the way it is.
Again rotation is inclining something continously.It is the same phenomenon , just seen from two different angles.
EULERS SPIRAL
|
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ABOUT
Then one day....a miracle happened,....while applying a certain formula again and again.....I came to know its significance. Slowly and steadily....other equations also started clicking. I got to see a strong relationship between Maths and the Physics it was pointing towards. They both were the same. Maths was just an easy language to express a physical phenomenon. And a bit more to that in the sense that it could even predict the behaviour of a certain physical phenomenon. Equations now as if came to life.
Every equation now had as if something to say. A burning urge to share these things with the world aflamed within me. So that no one has to give up the subject that he or she loves the most.
Every equation now had as if something to say. A burning urge to share these things with the world aflamed within me. So that no one has to give up the subject that he or she loves the most.
The book on visualizing maths thus got written as a sprout of inspiration. The blog followed. Both these are dedicated to you and all such similar minds searching for answers.
Visualizing Math & Physics is a blog dedicated to you and all such similar minds searching for answers.
CONTACT
binnoypanicker@gmail.com
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ReplyDeleteBinnoy.
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