|||| Curl basically is a measure of how much a field of force can turn you or spin you about its axis or about your axis.
|||| Obviously this ability depends upon how far you are from the center of the center of the force field (where usually the force field is the strongest.
| Example 2: A whirlpool |
Example 1: A hurricane
|||| When a whirlpool turns, the further you are from the center of the whirlpool, the lesser is the ability of the water involved to make you spin.
|||| So now you know, why always a partial derivative of x,y and z comes in the equations of curl.
|||| So now you know, why always a partial derivative of x,y and z comes in the equations of curl.
  |
| Havent you wondered why the cross 'X' sign comes in the formula of Curl ? |
|||| These partial derivatives just trace the path of how much the influence of the curling force or spinning force reduce as you move along the x-axis or y-axis or z-axis away from the centre (usually the source) of the curling whirlpool.
THE CROSS IN THE CURL.
|||| Just look carefully into the symbol of the curl, there is a cross in it which looks like 'X'. It comes in the cross-product as well as in the a the Curl.
|||| This Cross is similar to the cross on which Jesus was crucified.
|||| A Cross, needless to say is made of two bars which are fixed perpendicular to each other.
|||| Similarly, the 'X' symbol comes in maths whenever two forces produce maximum impact when kept perpendicular to each other.
But why does the Cross Come in the curl?
|||| The value of Curl is maximum if you are perpendicular to the direction of the flow of field.
|||| Putting in laymans terms, imagine a whirlpool and imagine a plank of wood caught in in. Now just ask yourself, how should the plank be placed so that it collects maximum energy of the whirlpool to keep spinning.
|||| Obviously, when it is kept at 90 degree w.r.t the flow of the river, it is going to collect maximum force of the water.
 |
| Pardon me, I didnt get a plank of wood for the image to explain. I only got something far more bigger though. |
|||| Forget a plank of wood, even if you keep your hand in a column of swirling water kept in a bucket, if you want your hand to be hit by maximum force of the swirling water, you have to place your hand perpendicular to the force of the water.
Other Spectacular examples of Curl in nature and in daily-life
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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.
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- In other words, the tendency of the pool to make you rotate is a function of your distance from the centre of the whirlpool. If you were in a boat…your boat would not only revolve, but also rotate about itself. This is because the water is hitting your boat strong on one side and weaker on the other side. This causes your boat to Curl.
- Now if the water was to push your boat uniformly in all directions,…..would your boat rotate around its own axis? No right. Because now Curl is absent.
- A better name for curl would be ‘spin’. It is the ability of a field to make you (or another object) spin.
ANALOGY 1
A ball left down a wash-basin will keep spinning around the centre of the basin. The closer it is to the centre, the faster it spins (i.e its spin is a function of its position)
Analogy 2
NASA once sent a probe to Venus initially, used the gravity of Venus to spin the probe around Venus itself, and when it gained sufficient momentum, targeted the probe towards Jupiter. The probe happily reached Jupiter on this free ride.
....AND........By the way......according to Einstein....massive objects create a curl in the surrounding space-time continnum leading to other objects like other smaller planets or satellites to follow a curved path....a curl.
 |
| Massive objects like the sun cause the surrounding space-time to curl thus causing any object to follow a curled orbit |
Analogy 3
SIGNIFICANCE OF i, j & k terms in CURL
1] Curl is a measure of how much an object can make you rotate or spin.
2] It varies as you move away from the centre (source) and so we have the partial derivatives along the x, y and z axises.
3] The 'X' sign in Curl is an indicator that the effect of Curl is maximum when the object is kept perpendicular to the flow of the continously curling field.
NOW ABOUT THE SIGNIFICANCE OF THE i, j and k TERMS IN THE EQUATIONS OF THE CURL.
|||| The i term or i plane lets say indicates revolution about the horizontal plane. That means the object is experiencing a force which is making it revolve in the horizontal plane (much like a planet revolving around the sun or a satellite moving along the equator of the earth)
|||| So the j term or j plane indicates revolution about the vertical plane. That means the object is experiencing a force which is making it revolve slightly vertically also. Imagine a satellite moving along an inclined orbit around the earth.
|||| The k term or k plane means now the object is rotating about its own axis also. So now our satellite is moving along an inclined path ( i.e slightly horizontally and slightly vertically simultaneously) and at the same time rotating about its own axis.
So........Basically
|| Only the i term means only horizontal curl or horizontal revolution.
|| Only the j term means only vertical curl or vertical revolution.
|| An i + j term would mean....horizontal as well as vertical revolution taking place simultaneously (inclined revolution).
|| And and i + j + k term would mean inclined revolution path plus rotation of the object about its axis.
|| Only k term might mean that the object is rotating about its axis,......but is not revolving about anything
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OTHER USERFUL IMAGES AND SITES UPON CURL
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| CURL AND VORTEX BY THE WAY ARE ONE AND THE SAME |
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Thank for your words. Keeps me inspired to dedicate time to this blog.
DeleteIts very synchronous of you as I was myself working on a new post to show the significance of i,j and k in the equations of Curl (On how they just mean different planes on rotation).
Binnoy
Very nice, it's exactly the kind of explanation I was looking for. Thank you very much!
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ReplyDeleteGreat !!!
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