Question about the Delta and Nabla symbols

[SemM]

Hi, in some books the $\nabla$ symbol is used for the Laplacian $\frac{d}{dx^2}+\frac{d}{dy^2}+\frac{d}{dz^2}$ while others use the $\Delta$ symbol for this. What is the correct custom for this usage?

 

[BvU]

$\nabla$ is used for something else:$$ \nabla = \hat \imath {\partial \over \partial x} + \hat \jmath {\partial \over \partial y} + \hat k {\partial \over \partial z} $$ https://en.wikipedia.org/wiki/Nabla_symbol#Modern_uses

And $\Delta = \nabla\cdot\nabla$

 

[Mark44]

Hi, in some books the $\nabla$ symbol is used for the Laplacian $\frac{d}{dx^2}+\frac{d}{dy^2}+\frac{d}{dz^2}$ while others use the $\Delta$ symbol for this. What is the correct custom for this usage?

It seems to me that the usual usage is $\Delta = \frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2}$. Notice that these are all second partials, not second derivatives, as you wrote.

 

[SemM]

It seems to me that the usual usage is $\Delta = \frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2}$. Notice that these are all second partials, not second derivatives, as you wrote.

Sound good! Is $\nabla$ related to a vector field, rather then a function?

and at last, what is $\nabla^2$?

 

[PeroK]

Sound good! Is $\nabla$ related to a vector field, rather then a function?

and at last, what is $\nabla^2$?

Calculus is a prerequisite for any physics. There is a complete calculus course (I, II and III) online here:

http://tutorial.math.lamar.edu/Classes/CalcIII/GradientVectorTangentPlane.aspx

 

[SemM]

Calculus is a prerequisite for any physics. There is a complete calculus course (I, II and III) online here:

http://tutorial.math.lamar.edu/Classes/CalcIII/GradientVectorTangentPlane.aspx

I had that too, 20 years ago...

 

[PeroK]

I had that too, 20 years ago...

Well, you need to revise it. You not only need to know what these symbols mean but at least have some experience and confidence of working with them.

 

[SemM]

Well, you need to revise it. You not only need to know what these symbols mean but at least have some experience and confidence of working with them.

Hi Perok, I did a course in Advanced Maths at my Univ 2 years ago, with some of this, and it worked out well, I got 5 out of 6 grades. The problem is that, I am lazy.

 

[Mark44]

The problem is that, I am lazy.

Something you'll need to address. Since you apparently are studying fairly advanced physics, you need to put in some effort at reviewing (or learning) the mathematics required. You shouldn't expect us to do all the legwork.

 

[BvU]

Lazy buggers google vector calculus and Laplacian to freshen up their memory (in my case it was 45 years ago).
A lot of textbooks^* (Jackson) also have an appendix or inside cover with useful identities.

* You know: these stacks of paper, bound with carboard covers

 

[Orodruin]

You shouldn't expect us to do all the legwork.

Not only should this not be expected, it is an impossibility. We cannot learn maths for him. Without the appropriate background knowledge in mathematics, there is absolutely no way to understand physics properly. If he cannot overcome his laziness, he will not learn physics. It is that simple.

 

[SemM]

Something you'll need to address. Since you apparently are studying fairly advanced physics, you need to put in some effort at reviewing (or learning) the mathematics required. You shouldn't expect us to do all the legwork.

I am not. I read something like 3 hours a day, I am just writing that I am lazy as a person, but it doesn't mean I am lazy in this context. In fact, if you look at all my posts in QM and Algebra, they show question on advanced topics without having understood the basics. That doesn't mean I haven't tried.

 

[SemM]

Lazy buggers google vector calculus and Laplacian to freshen up their memory (in my case it was 45 years ago).
A lot of textbooks^* (Jackson) also have an appendix or inside cover with useful identities.

* You know: these stacks of paper, bound with carboard covers

:D

 

[SemM]

Not only should this not be expected, it is an impossibility. We cannot learn maths for him. Without the appropriate background knowledge in mathematics, there is absolutely no way to understand physics properly. If he cannot overcome his laziness, he will not learn physics. It is that simple.

Again, its not about laziness in reading, its about the personality. I read physics around3 hrs pr day, so I am not lazy on that, I am lazy in the learning process. Or maybe I should use the word slow.

Do not overreact on this term laziness.

 

[Orodruin]

In fact, if you look at all my posts in QM and Algebra, they show question on advanced topics without having understood the basics. That doesn't mean I haven't tried.

This begs the question: Why are you trying to learn advanced topics without mastering the basics? Without understanding the basics, you have very little chance of understanding those advanced topics. I would suggest that you first try again to master the basics, if you do not you will not be able to handle the advanced topics.

 

[SemM]

This begs the question: Why are you trying to learn advanced topics without mastering the basics? Without understanding the basics, you have very little chance of understanding those advanced topics. I would suggest that you first try again to master the basics, if you do not you will not be able to handle the advanced topics.

If you want a full answer to this, I must tell you what my teacher in Physical Chemstry in the Engineering Degree in Chem gave me on the exam and told me:

"Sergio, you have received 2.1 in a scale of 4 to 1, when 1 is the highest score, and you solved all the difficult exercises but failed on the easy ones, why?"

I said, the difficult ones are more interesting.

 

[Orodruin]

If you want a full answer to this, I must tell you what my teacher in Physical Chemstry in the Engineering Degree in Chem gave me on the exam and told me:

"Sergio, you have received 2.1 in a scale of 4 to 1, when 1 is the highest score, and you solved all the difficult exercises but failed on the easy ones, why?"

I said, the difficult ones are more interesting.

This does not mean that you will get away with the same thing in all subjects. In order to understand the answers to many of the questions you have been posting you really need to understand the basics. If you skip the ”easy” ones because they are trivial to you, fine, but based on what you have said here and in other threads, it does not appear to be the case. It is also not the same thing to reproduce things in an exam as it is to understand the subject on a fundamental level.

 

[SemM]

This does not mean that you will get away with the same thing in all subjects. In order to understand the answers to many of the questions you have been posting you really need to understand the basics. If you skip the ”easy” ones because they are trivial to you, fine, but based on what you have said here and in other threads, it does not appear to be the case. It is also not the same thing to reproduce things in an exam as it is to understand the subject on a fundamental level.

I take that the best description of what I tried to answer PerOK is that I may be more eclectic. But I do agree completely with you and all the others in this thread and others, that that may not work out so well as in other fields or in exams. Thanks for your advice. I will implement that in my learning process!

Thanks Orodruin

 

[Mark44]

I am lazy in the learning process.

Or maybe I should use the word slow.

Those are two different things. For most people it's not possible to learn advanced physics and mathematics at a rapid clip.

 

[SemM]

Those are two different things. For most people it's not possible to learn advanced physics and mathematics at a rapid clip.

I am not really following the main stream instinctively, but as I said to Orodruin, I will try, at least in this realm, because I see repetitively, that except errors which do occur here and there, it has been preserved against nonsense and a lot of non-tangible and non-proofable theories, such as Bayesian mechanics, and I therefore have the deepest respect for both physics, and not to mention mathematics. But I must admit that when I read about linear spaces in the Fundamentals of the Theory of Operator Algebras by Kadison and Ringrose, chapter 1, my head explodes of boredom, but when I read chapter 2 in the same book "Basics of Hilbert spaces and Linear Operators" I enjoy it so much I forget to make dinner.

This all started when I was asked to proofread a renowned QM book for a professor, and I found 8 errors in the book in the first 2 weeks, but still after reading it after 7 years, I still don't really feel that I understand quantum mechanics like "one should", and I often have to go back and re-check, the Born condition, the Prescriptions.

However, the opposite occurs as well, for instance, just yesterday when reading that excellent chapter 2, I realized what a continuous functional space means, and that it MUST be closed if it is continuous. It took me 4 months, but that book wrote it in a way it fell together in just a moment.

 

[Mark44]

But I must admit that when I read about linear spaces in the Fundamentals of the Theory of Operator Algebras by Kadison and Ringrose, chapter 1, my head explodes of boredom

Does this mean that you understand all of the material in this chapter? From some of your posts I've seen, it doesn't seem that you have a clear understanding of "linear-ness" in linear spaces.

 

[SemM]

Does this mean that you understand all of the material in this chapter? From some of your posts I've seen, it doesn't seem that you have a clear understanding of "linear-ness" in linear spaces.

Absolutely not. The boredom (or maybe its not that either) comes from seeing that although it is a very simple point they are posing, they pose it in such a way it takes half-a-page of paths that reach any corner of a logical reasoning, before coming back to a very simple point. When it gets harder, then I understand why they use a whole page to prove it, and I get interested in the subject.