Note: I could have explained it in a formal way, but I did not because it would have been hell boring and less interesting if I had done otherwise.

The clause in bold makes me adopt one of the two available options:
1) Should I directly explain you the curl and the divergence by disregarding the rudimentary stuff required to understand the above topics?

2) Should I elucidate the basic stuff first and then move on to the subject-matter? In short, should I spoon-feed you by elaborating the rudimentary stuff first?

Let flip the coin...

IslamabadKid vs IslamabadKid's intuition...

Ramez Raja: "Heads or Tails?"
IslamabadKid: "Heads"

Ramez Raja while tossing the coin in the air: "Heads is the call..."
*coin dancing in the air while singing: Main uddi uddi javan Hawa de nul... touches the ground... settles on the ground*

Ramez Raja: "It's tails!!! IslamabadKid's intuition, what do you want to choose?"

IslamabadKid's intuition: "I want choose the option 2!"

So it is settled! IslamabadKid succumbed to his intuition, and ready to explain the rudimentary stuff first.
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The very first question that arises in one's mind is: What is so special about the vectors?

The answer to that is pretty straightforward, which is the "vector" itself.

The very next question that loomed up in one's mind is: Alright! Although that was obvious, what exactly is the vector?

Vector is the quantity that has a magnitude as well as the direction.

So you mean to say that anything that has some value associated to it and it is moving, heading or sets off in the specific path of its movement?


Yes.

Okay! If it is so, is there any general representation of that? This question I am asking because whenever I travel, I see few symbols on specific points along the road, pointing towards the specific path. Also, "seedha aagay jayein... jaatay jayein... jaatay jayein... phir saj'jay ho jayein... aagay chaltay jayein... aur phir khabbay ho jayein... wahan aap ko aik Rickshaw nazar aye ga... uss Rickshay walay say sahi raasta pata karein, kyun kay mujay bhi nahi pata keh apki destination kahan hay exactly!"

That seedha, khab'ba, saj'ja are the specific directions, and the distance(s) we cover is/are the magnitude(s) associated to that path(s).


The most obnoxious, but true and to-the-point, interpretations you have mentioned. As far as the only question residing in that whole essay, which was how to represent the vector, is concerned, you need to imagine an arrow - the real arrow, like:

x------>

That "x", which you stick in the rope of a bow, is the back of an arrow.
That ">", which can penetrate into one's body easily because of its pointy surface, is the front of an arrow.

Did you know: have you seen a real arrow in your life? If so, have you noticed one cool thing about it? If not, let me tell that.

If you look at the arrow from its front, you will see just a "dot" of the front pointy surface. If you look at the arrow from its back, you will only see the "cross"(x) of its rear end.

Whenever you see encircled "."(dot) on the paper or a book, it means the arrow is coming out of the paper; and if you see encircled "x"(cross) on the paper or a book, it means the arrow is going into the paper.

I hope it makes you remember the encircled "." and encircled "x" convection.
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Now imagine your room's one wall out of four is firing vectors instead of bullets. *lol* The condition is that the vectors do not cross each other, meaning they are being fired in the straight lines like:

--> -----> --------> --------->
--> -----> --------> --------->
--> -----> --------> --------->
--> -----> --------> --------->
--> -----> --------> --------->
--> -----> --------> --------->
--> -----> --------> --------->
(Vectors* travelling along x-axis)
*Vector field(shall discuss later)

As you can see, the vectors that are being fired by your left wall are heading along the x-axis, not y-axis or z-axis.

What if the vectors are fired by the ground of your room in upward direction?
It means they(vectors) will be heading along the z-axis, not x-axis or y-axis.

What if vectors are fired by the wall behind you in forward direction?
It means those vectors will be heading along the y-axis, not x-axis or z-axis.

What we have learned from above discussion is that the vector field* can be along any direction.

Now if you want to know of the change in vector field(vectors in your room) happening at any point in the room, all you need to do is to find the change along x-axis by ignoring y-axis and z-axis, to find the change along y-axis by ignoring x-axis and z-axis, and to find the change along z-axis by ignoring x-axis and y-axis.

Add all of them, you will get the result.

Now in mathematics/Physics, we use "partial fractions" so as to encounter that ignoring part of the above statements. What do I mean by that? It means the following:

d/dx i = change along x-axis by ignoring y-axis and z-axis.
d/dy j = change along y-axis by ignoring x-axis and z-axis.
d/dz k = change along z-axis by ignoring x-axis and y-axis.

Please make sure that "d" is CURVY. I cannot type the curvy "d" in this forum as it does not support the mathematical symbols.

Also, i,j, and k represents the unit vectors along x, y and z axis respectively. Furthermore, there will be "caps" on top of i, j and k.

Now the overall change of vector field,which is called the GRADIENT, can be found by adding all of the above notions, meaning:

d/dx i + d/dy j + d/dz k = Overall change of vector field = Gradient of vector field

Gradient is represented by the symbol "∇".

Please understand all of the above points first, and tell me what is unclear to you.

Once you understand all of the above discussion, we shall move on to the next, important topics:
1) Vector fields,
2) Divergence,
3) Curl.

All of the above topics especially (2) and (3) cannot be explained without the knowledge of gradients, vectors, and vector fields.

I shall use the aforementioned jargon over and over again while explaining Divergence and the Curl. Therefore, please understand that first.

Do let me know when you completely comprehend everything mentioned in this post!

If you face any problem, do mention that too.

P.S: 30-35% course of vectors is explained in this post. <--- just to motivate you!

*Suspense* lol (to-be-continued)

Yes, i got it in a humorous way. Sorry to say, i will only seek guidance in the most difficult topics. Did not want to disturb your pace because of me. Thanks

First, I come here in my spare time.
Second, explaining something does not disturb my pace; it makes me revise my concepts.
Third, humorous way is one way to entertain oneself after a hectic day.
Fourth, getting something "economical" out of that humorous way is its another benefit.
Fifth, and above all, you cannot understand "curl and divergence" without the knowledge of gradients, as ∇.f = divergence; ∇xf = curl.

Regards.

Continuation of the last post, of which theme was rallying around the following topics:
1) Vectors and their conventions;
2) Change in any direction;
3) Mathematical representation of that change - aka Gradients;
4) A slight introduction to the vector field.

Now let me continue from the topic I left off the discussion, which was the vector field.

The question is: What exactly is the vector field?

But hey wait! You know what a vector is, but what about the "field?"

In simple terminology, the word "field" can be defined as an area of an open land.

The very next question is: Okay! If that is the definition of a field, how would you connect its definition to the notion "vector field?"

The simple interpretation you can get by considering the definition of the field is that we have the area, which is filled with vectors.

The formal definition of the vector field is:

Let us dissect the aforementioned definition!

1) Space: You may call that the space is tantamount to your room's space. In-short, vacant area inside the four-walled room.

Let me relate it to the real world!

For instance, your parents have bought a new house, and now you are ready to transfer each and everything of yours into your room.

At first glance, you would say, "Oh! This is a big room! It has a lot of space in its vicinity. Thank you Baba!"

*Now let's suppose that you have a weird -in others' frame of reference- habit of buying at least one book every week, meaning you have a lot of book that you need to accommodate into your room.*

Now you are "transferring" all of your stuff into your new room! After ten minutes, you will realise that you do not have enough space in your room for your books. You reaction might be:*sigh* + "Awein he khush tha/the main kamra dekh k! Now I do not have space for my books."

Now let me take "one" book and map it using Mathematics and Physics.

Every book has width, height, and depth, meaning it is just a 3-dimensional "mass" taking the space, which is width*height*depth(volume).

Why have I take 3-dimensions? because it encompasses the lower dimensions as well - which are 1-dimension and 2-dimensions; therefore, I do not want to go deep into the lower dimensions, for the intuition behind them is fairly easy. Later on, I shall take 2-dimensional area for ease.

2) At each point: Meaning every point in the space; like:


Suppose above is your room, and the red dots represents the "points" in the "space."

NOTE: There are infinite points; I have shown very few.

3)Vector quantity: I have explained it in detail in my last post. I am assuming that you have understood that concept.

Now let us connect the dots, which in this case are three mentioned above;

Every point in the space is represented as a vector. Or the other way to elucidate is that a vector - a quantity that has a magnitude as well as direction- is assigned to each and every point in the space.

Like:


Again, I have drawn very few vectors associated with the points in the space.

Having said that, the tragic story of vector field comes to its end.

It's ending -happy or sad- has been left to your imagination.
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Let me show you few of the vector fields:



The above picture shows you the 3-dimensional "space" populated with the "vectors" assigned to few "points." You may relate it to your room, and imagine the vectors flying around you like mini-aeroplanes. You can imagine really cute scenario before going to sleep about 3-dimensional space and the vector field.



The above picture shows you the 2-dimensional "space" populated with the "vectors" assigned to few "points." You may relate it with A-4 paper, and imagine that vectors are moving on that paper.

y
^
|

---> ----> ----> ---> --> ->

->x

The above stream of vectors, imagine, are along x-axis only. <-- One dimensional vector field.

(The end)


Next topic: Divergence(Finally)! I hope you get something out of the post above regarding vector fields.

Cheerio!