Book Div, grad, curl, and all that thing / Chapter III - I find it difficult to follow

[MaxMath]

I thought this book should help me better understand these concepts, which I sort of did but always felt the existence of gaps somewhere. With high expectations, I find chapter III difficult to follow (starting from ebook p 72 / page number 63). The author appears to take an approach based more on intuition (sacrificing mathematic rigour), which is not necessarily bad. However, this makes it difficult for me to follow its flow. This is probably due to the position I'm in; that is, I have some degree of intuition of all of these concepts, but probably will benefit from an exposition with rigour to fill the gaps.

If you are good at this subject and have a moment to spare, can you please share your comments on this chapter and the author's handling of the subject?

Not much prep is needed except that it's the 'convention' of this book to use [imath]F_x[/imath] to mean the [imath]x[/imath] component of a vector function [imath]F(x, y, z)[/imath], rather than the partial derivative of [imath]F[/imath].

Thank you.

 

[mmm4444bot]

The author appears to take an approach based more on intuition

Hello. Can you provide a specific instance or two? Is your assessment related to physics used as examples, lack of proofs, expository style (eg: skipping steps), something else? Members might be able to suggest different texts, if they knew something about your learning preferences.
[imath]\;[/imath]

 

[mario99]

I thought this book should help me better understand these concepts, which I sort of did but always felt the existence of gaps somewhere. With high expectations, I find chapter III difficult to follow (starting from ebook p 72 / page number 63). The author appears to take an approach based more on intuition (sacrificing mathematic rigour), which is not necessarily bad. However, this makes it difficult for me to follow its flow. This is probably due to the position I'm in; that is, I have some degree of intuition of all of these concepts, but probably will benefit from an exposition with rigour to fill the gaps.

If you are good at this subject and have a moment to spare, can you please share your comments on this chapter and the author's handling of the subject?

Not much prep is needed except that it's the 'convention' of this book to use [imath]F_x[/imath] to mean the [imath]x[/imath] component of a vector function [imath]F(x, y, z)[/imath], rather than the partial derivative of [imath]F[/imath].

Thank you.

I finished my Calculus III a few years ago. I would be so glad to discuss any specific problem with you.

 

[MaxMath]

Thanks both. Will come back with more specific questions or examples from the book.

In the meantime, if anyone has suggestion on materials related to this subject, aiming at bridging mathematics and intuition, AND with rigour, I'm all ears.

 

[Cubist]

I found the book "Further Engineering Mathematics" by K A Stroud useful. You'll probably find it in most university engineering libraries therefore I recommend thumbing through it before borrowing to see if it suits your needs.

What is your intuitive understanding of each of these operators? What does each operate on and return ("scalar field" or "vector field")?

 

[MaxMath]

I found the book "Further Engineering Mathematics" by K A Stroud useful. You'll probably find it in most university engineering libraries therefore I recommend thumbing through it before borrowing to see if it suits your needs.

What is your intuitive understanding of each of these operators? What does each operate on and return ("scalar field" or "vector field")?

Thank you. I don't have access to uni library for now; I left uni donkey years ago.

This book is on Internet Archive. Unfortunately however, presumably due to a law case that is going on, it's not up for borrow. Nonetheless, I found several other books with a slightly different title, Advanced Engineering Mathematics.

I think I already have reasonably good intuition of all of these concepts, including the types of the domain/codomain (scalar vs vector) for each of these. I wanted a better understanding of curl in particular, and this is probably interrelated to the problem of rotation of rigid body, which again shakes hands with my recent quest into the phenomenon of tides, and another interesting topic of precession.

One of these books I found is already interesting. One specific question that came up (tho this is digressing from the original question, but certainly related) --


This theorem joins curl and rigid body rotation. But it puzzles me somewhat. In my understanding, curl of a vector field is also a vector field. This aspect is however opaque in this theorem. I think it's safe say, on the basis of it, that the curl along the rotating axis of the rigid body, the curl of the velocity field is a vector along this axis with magnitude twice the angular speed of the rotation. It's not clear to me, at least from the expression of the theorem, whether the curls at other locations are also the same (such as on the edge of rotating disc). I believe it is, based on the derivation of this theorem, giving a curl (field) that is a constant.

And if so, it's interesting (and somehow counterintuitive) to me. This shows somehow the nature of rotation.

[Some more context: In my quest into the phenomenon of tides, I tend to think rotation is absolute, while - as we all know - translational movement is relative. And in the process, I 'discovered' some nuances of what we are commonly taught, such as the Moon rotates around the Earth. This is frustratingly inaccurate; the truth is, it's both the Earth and the Moon as a combination, rotating their common centroid, albeit being close to, even under the surface of, the Earth. But nonetheless, we should not say the Moon rotates around the Earth, as it the Earth was still or only translationally moving, or even rotating around the Sun, which again is similarly inaccurate, in my opinion.]

I will have more reads of the subject book, and I guess to achieve what I wanted, the only way is probably find all the related materials that I can find and join them together to see if that helps. It will require, of course, more study of the interpreted subjects that interest me.

This is not urgent but rather a part of where curiosity drives me to.

 

[Cubist]

This theorem joins curl and rigid body rotation. But it puzzles me somewhat. In my understanding, curl of a vector field is also a vector field. This aspect is however opaque in this theorem. I think it's safe say, on the basis of it, that the curl along the rotating axis of the rigid body, the curl of the velocity field is a vector along this axis with magnitude twice the angular speed of the rotation. It's not clear to me, at least from the expression of the theorem, whether the curls at other locations are also the same (such as on the edge of rotating disc). I believe it is, based on the derivation of this theorem, giving a curl (field) that is a constant.

And if so, it's interesting (and somehow counterintuitive) to me. This shows somehow the nature of rotation.

You might find it useful to have a play with this 2d field visualization tool that I just found:-
https://www.geogebra.org/m/cXgNb58T

If you enter (-y, x) as the field then you obtain a rotation:-



And then you can ask Wolfram to perform the curl calculation for you:-
https://www.wolframalpha.com/input?i=curl[-y,x,0]

curl[ -y, x, 0 ] = (0, 0, 2)

The farther each point is from the origin the velocity gets bigger. BUT, the whole plane performs a complete rotation (each point returning to it's starting point) in exactly the same time. Every point rotates at the same rate. The velocity is bigger farther away (from the origin) because it has more distance to cover.

EDIT: Perhaps think of a smiley face emoji placed somewhere on the plane. As the plane rotates, the emoji returns to its starting position and it would have rotated once. Now imagine another emoji placed somewhere else on the plane. Both emojis will rotate the same amount as the plane rotates. Try it with some stickers on a piece of paper that you turn by hand!

EDIT2: Note that the curl operator can't tell you where the centre of a rotating field is. It just tells you how each point is rotating

his is frustratingly inaccurate; the truth is, it's both the Earth and the Moon as a combination, rotating their common centroid,

Indeed, and this helps to explain why there are two tidal bulges every day (and not just one as you might expect if the Earth didn't move during the dance of the Earth and Moon). Google "why are there two tides a day" (sorry if you are already aware of this )

When learning about a physical subject it's often easier to start with a very basic (mathematical) model. Then, as understanding improves, a student can be presented with an improved model which better represents a physical situation and gives more accurate results. This is a good thing in my opinion and different levels of model exist for all kinds of things:- electronic components, macro economics, etc. Not everyone is capable of understanding the most accurate models that we know of (which might not be perfect anyway) !

 

[MaxMath]

You might find it useful to have a play with this 2d field visualization tool that I just found:-
https://www.geogebra.org/m/cXgNb58T

If you enter (-y, x) as the field then you obtain a rotation:-

View attachment 38246

And then you can ask Wolfram to perform the curl calculation for you:-
https://www.wolframalpha.com/input?i=curl[-y,x,0]

curl[ -y, x, 0 ] = (0, 0, 2)

The farther each point is from the origin the velocity gets bigger. BUT, the whole plane performs a complete rotation (each point returning to it's starting point) in exactly the same time. Every point rotates at the same rate. The velocity is bigger farther away (from the origin) because it has more distance to cover.

EDIT: Perhaps think of a smiley face emoji placed somewhere on the plane. As the plane rotates, the emoji returns to its starting position and it would have rotated once. Now imagine another emoji placed somewhere else on the plane. Both emojis will rotate the same amount as the plane rotates. Try it with some stickers on a piece of paper that you turn by hand!

EDIT2: Note that the curl operator can't tell you where the centre of a rotating field is. It just tells you how each point is rotating

Very good information out there. Thanks.

Re you EDITs, that to me is the hard part -- every point is rotating at the same rate (which is ok), then there is no information as of with respect to what point it's rotating! Also by intuition with the help of , it seems to me that the movement of any point off the centre is a superposition of two rotations!

Indeed, and this helps to explain why there are two tidal bulges every day (and not just one as you might expect if the Earth didn't move during the dance of the Earth and Moon). Google "why are there two tides a day" (sorry if you are already aware of this )

Exactly! I now understand this pretty well, after having undertook a quantitative analysis of the tidal force (or to be more precise, the tide-causing force). Here are some snapshots of a spreadsheet I made showing, to scale (except the distance between Earth/Moon, shown as broken lines), the components and their combined force, including direction and magnitude.

I need yet to improve this to illustrate the two 'bulges', and potentially, to contrive a chart showing the timing of hi/low tides and periods etc for a given place in the ocean or along the coast (based on simplified model).

When learning about a physical subject it's often easier to start with a very basic (mathematical) model. Then, as understanding improves, a student can be presented with an improved model which better represents a physical situation and gives more accurate results. This is a good thing in my opinion and different levels of model exist for all kinds of things:- electronic components, macro economics, etc. Not everyone is capable of understanding the most accurate models that we know of (which might not be perfect anyway) !

Totally understand. But this bit is really critical. Because in the universe, from the perspective of stars/planets/satellites, there is nothing for you to hold on to so to allow something to orbit around you without dragging you into the 'dance'!

[By the way, apologies for my too many typos in my previous post.]

 

[Cubist]

Very good information out there. Thanks.

You are welcome!

Re you EDITs, that to me is the hard part -- every point is rotating at the same rate (which is ok), then there is no information as of with respect to what point it's rotating! Also by intuition with the help of , it seems to me that the movement of any point off the centre is a superposition of two rotations!

The rotation indicated by curl is JUST around the point in question (because we can plug in specific x and y values into curl's result). curl can be used on very complicated vector fields which have no single centre of rotation (perhaps think about a fluid flow around an object). Saying this, I have never used curl in any practical sense myself

If you're looking for a method to determine the centre of a planar rotation then curl probably isn't the best function to use. I do have some ideas about how I would go about this...

Exactly! I now understand this pretty well, after having undertook a quantitative analysis of the tidal force (or to be more precise, the tide-causing force). Here are some snapshots of a spreadsheet I made showing, to scale (except the distance between Earth/Moon, shown as broken lines), the components and their combined force, including direction and magnitude.

Your work on the tidal force looks impressive! You must know a lot more about this than I do