How do we define and calculate divergence in vector fields?

[Urmi Roy]

That's fine with me,but you see,I have been rushing a little since I have exams every now and then...after this I have some points to get cleared about line integrals,surface integral and the theorems of vector calculus that I have just read through from Kreyzig(I don't have sound understanding of these..and believe me my teacher's can't help me!)...so I do have a little problem with time.
Will you guide me through what's remaining of vector calculus?...then I can take it easy and slowly,and rely on the fact that at the end of a couple of days, I'll be thorough in my understanding of vector calculus(I'll try not to bother you too much after that.)

 

[Urmi Roy]

Everyone forgotten me? :(

 

[Urmi Roy]

Okay,since I don't have much time left before my exams,I need to get my last set of questions cleared on curl...please please please help!

1. Is it possible to have Curl of velocity? Intuitively,what would that be?

2. I have noticed that the del operator used to define divergence and gradient fit in perfectly with the definition of these quantities...how does the determinant formulation of curl fit in with the true meaning of curl?...why do we use this determinant formulation?

3. In wikipedia,they interpret curl with the help of a ball with rough edges,and how the fluid moving past it makes it rotate...the magnitude of the rotational velocity is the value of curl...but this cannot be applied to any other vector(all vectors wouldn't make a ball rotate,e.g. displacement vector).

4.Talking of displacement vector,what is the curl of this vector...how would we interpret it intuitively?

5."The curl of a vector field F at a point is defined in terms of its projection onto various lines through the point. If is any unit vector, the projection of the curl of F onto is defined to be the limiting value of a closed line integral in a plane orthogonal to as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed."- this is from wikipedia...what does it mean?

My last and perhaps most important question:
The curl is defined as the 'microscopic circulation per unit area' (ref: http://www.math.umn.edu/~nykamp/m2374/readings/circperarea/) ...how does this definition tally with the usual way of defining curl that I mentioned earlier in the post?

 

[elibj123]

1. It will tell you about the angular velocity/acceleration at that point, but that's not very accurate.

2. If we jump ahead to 5 (which I'll explain later) and look at its definition, we can now define the components of the curl field along the xyz axes. Since the shape of the infinitesemal loop is not important (assuming everything is smooth enough), I would suggest taking a rectangular with sides parallel to the different axes (while computing you will have three of this loops). Calculating the circulation density:

curl(F) _{x}=lim \frac{\oint_{C_{yz}}\vec{F}\vec{dr}}{A_{yz}}
(C_{yz} is a rectangular loop parallel to the yz plain)

With this specific shape, will give you exactly the definitions of the derivatives. Summing all the components, you will recognize the structure of a cross product, or as you said, the determinant formulation.

3 & 4. This is just an explanation with analogy. However I don't see how a displacement vector can play a role of a vector field. A displacement vector is usually a function of t. With curl we are talking about functions of (x,y,z).

5. The projection of the curl against a specific direction, will tell you how much the field circulates around that direction at some point.

 

[Urmi Roy]

1. It will tell you about the angular velocity/acceleration at that point, but that's not very accurate.

So it doesn't really have any physical existence,right?

2. If we jump ahead to 5 (which I'll explain later) and look at its definition, we can now define the components of the curl field along the xyz axes. Since the shape of the infinitesemal loop is not important (assuming everything is smooth enough), I would suggest taking a rectangular with sides parallel to the different axes (while computing you will have three of this loops). Calculating the circulation density:

curl(F) _{x}=lim \frac{\oint_{C_{yz}}\vec{F}\vec{dr}}{A_{yz}}
(C_{yz} is a rectangular loop parallel to the yz plain)

With this specific shape, will give you exactly the definitions of the derivatives. Summing all the components, you will recognize the structure of a cross product, or as you said, the determinant formulation.

Thanks,I've understood this.

A displacement vector is usually a function of t. With curl we are talking about functions of (x,y,z).

I should have taken this into account earlier!

5. The projection of the curl against a specific direction, will tell you how much the field circulates around that direction at some point.

Lastly,from what I get (especially from your answer to 3 and 4,)curl doesn't really have to make something (like a ball or a paddle) rotate...it's just a measure of how much the vector arrows are twisting,right?

 

[Urmi Roy]

Please forgive me for extending even further...just my last two questions..

1. When we calculate curl,we reduce the area over which the path integral is being integrated to a single point--however,due to this, the magnitude of curl should always be very small,as the path integral over an area as small al that would be near to zero!
Am I saying something wrong?

2.Direction of curl is defined as the direction of maximum rotation...what doe the 'maximum' imply?

 

[Lord Crc]

1. When we calculate curl,we reduce the area over which the path integral is being integrated to a single point--however,due to this, the magnitude of curl should always be very small,as the path integral over an area as small al that would be near to zero!

But you divide by the (near-zero) area as well. This is similar to how the derivative is defined. The difference f(x+h)-f(x) will also be nearly zero (for continuous functions) as h goes to zero, but you divide by h, and so you get something else for the limit.