Curl of a vector

Mathslover123

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Many problems are solved by this approach in my book.But i dont understand how this relation is true and derived . It has replaced three components into summation of single one.So help me understand this.I have pointed an arrow where i have trouble understanding .
 

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Dr.Peterson

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I've never seen this notation, but it looks to me as if the summation means "the sum over all basis vectors i and variables x", as if \(\displaystyle \vec{i}\), \(\displaystyle \vec{j}\), and \(\displaystyle \vec{k}\) were subscripted vectors \(\displaystyle \vec{i_1}\), \(\displaystyle \vec{i_2}\), and \(\displaystyle \vec{i_3}\) respectively, and variables x, y, and z were subscripted variables \(\displaystyle x_1\), \(\displaystyle x_2\), and \(\displaystyle x_3\).

Has the book ever explained such a notation? They need to have done so before using it.
 

Mathslover123

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I've never seen this notation, but it looks to me as if the summation means "the sum over all basis vectors i and variables x", as if \(\displaystyle \vec{i}\), \(\displaystyle \vec{j}\), and \(\displaystyle \vec{k}\) were subscripted vectors \(\displaystyle \vec{i_1}\), \(\displaystyle \vec{i_2}\), and \(\displaystyle \vec{i_3}\) respectively, and variables x, y, and z were subscripted variables \(\displaystyle x_1\), \(\displaystyle x_2\), and \(\displaystyle x_3\).

Has the book ever explained such a notation? They need to have done so before using it.
Not mentioned anywhere.Every problem is solved by using this relation in the beginning.
 

pka

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Not mentioned anywhere.Every problem is solved by using this relation in the beginning.
What sort of answer is that? I second everything Prof. Peterson said.
I have never seen any vector-notation as used in your post. What is the term \(\displaystyle \bf{r}~?\)
At times it seems to be both number & vector in the same term. So which is it?
Moreover, the cross product of two vectors is a vector. In your post don't you have a vector divided by a vector?
 

Dr.Peterson

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Perhaps it will help if we know a little more about the book. What book is it (title and subject), and what prerequisite knowledge does it assume from the start? Is there any introductory section that might at least "remind" you of notation (to make sure you will understand what they say)?

I'm guessing that it is not a math text, but physics or something ...
 

Mathslover123

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I apologize for not mentioning. Its a mathematics problem in a chapter Gradient , divergence and curl of vector. Is there any other way to solve this kind of problem ?
 

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topsquark

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I'm guessing that it is not a math text, but physics or something ...
Physics does tend to make some "obvious" notations more difficult but I've never seen the notation in the OP either. In fact I didn't run across this type of problem until late in my Undergrad. At that point even the Physics books start getting it mostly right.

-Dan
 

pka

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Physics does tend to make some "obvious" notations more difficult but I've never seen the notation in the OP either. In fact I didn't run across this type of problem until late in my Undergrad. At that point even the Physics books start getting it mostly right. -Dan
Dan, I am glad to see a physicist post that. Having taught vector analysis many times, I have never seen such poor notation in print. not ever.
But much beyond even that, why do you think Mathslover123 refuses to answer question as to what say how \(\displaystyle {\bf{r}}~\&~\vec{r}\) differ.
In one case \(\displaystyle \frac{1}{\bf{r}}\) is used with a vector. What do they mean?
 

Dr.Peterson

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It looks to me like they use \(\displaystyle r\) without an arrow to mean the magnitude of the vector \(\displaystyle \vec{r}\). That's not an entirely unreasonable notation, if they state it somewhere and are consistent, though a standard notation would seem better; I would really like to see the beginning of the book in order to make a more accurate judgment of it.

But the original point of the thread is not to criticize the book (much as we might like to), but to determine what it means. I think we've figured that out.

The new question is, "Is there any other way to solve this kind of problem?" That's a little vague; I'm not sure if it refers to avoiding the summations in the original example, or something about the newly shown page. Any ideas?
 

topsquark

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In one case \(\displaystyle \frac{1}{\bf{r}}\) is used with a vector. What do they mean?
Typically \(\displaystyle r \equiv | \textbf{r} | \). It usually works out. Until you start using indices anyway. At that point I tend to have to use \(\displaystyle | \textbf{r} | \) just so I can keep track of the d*mned indices. (GR is a monster for me that way. I know what the indices do but I tend to lose sight of the forest for the trees.)

-Dan
 

pka

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It looks to me like they use \(\displaystyle r\) without an arrow to mean the magnitude of the vector \(\displaystyle \vec{r}\). That's not an entirely unreasonable notation, if they state it somewhere and are consistent, though a standard notation would seem better; I would really like to see the beginning of the book in order to make a more accurate judgment of it.

But the original point of the thread is not to criticize the book (much as we might like to), but to determine what it means. I think we've figured that out.
I don't see that anyone was critical of any textbook. Moreover, I agree that it is not unreasonable notation to use \(\displaystyle {\bf{r}}=\|\vec{r}\|\) but for crying-out-loud why not say so after being asked at least twice?
 

Dr.Peterson

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Well, my own criticism of the book is that they seem to be using this notation (the summation without indices) without having ever defined it. I want to get myself past that!
 

Mathslover123

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Sincerely speaking i may have very poor knowledge about this very topic but i can say that 'r' without arrow head represents the magnitude,sir.And sir lets find a solution to these questions in any way we can rather than talking about authors mistake for not mentioning that very thing .
Sorry for the late response !
 

Dr.Peterson

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That's what I've been saying. But I asked you to clarify what you are now asking for. I think we dealt with the original question of what the summations mean. What is still needed? Be specific, to get us on track!

You asked, "Is there any other way to solve this kind of problem?" Which problem are you referring to, and what sort of other way are you hoping for?
 

Mathslover123

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Through my experience till now , i think that mathematics problems has number of ways to solve.By other ways , i mean other ways which does not involve replacing three components of divergence into summation of just 'i' component.
I think it can also be solved by first taking the cross product of a and r and dividing that by mag. Of r . Then finding its divergence.I think It would be quite lengthy.
I have posted a pic..Just help me by doing further steps in it .
 

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