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January 1st, 2014, 04:12 PM   #1
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Curl and divergence of units vectos

Hellow!

I'd like to know what results the curl and divergence of unit vectos bellow:

I just know that

?x = 0
?y = 0
?z = 0

?x = 0
?y = 0
?z = 0
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January 2nd, 2014, 06:27 AM   #2
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Re: Curl and divergence of units vectos

Quote:
Originally Posted by Jhenrique
Hellow!

I'd like to know what results the curl and divergence of unit vectos bellow:

I just know that

?x = 0
?y = 0
?z = 0

?x = 0
?y = 0
?z = 0
I'm not certain I understand your question or notation- and cannot see your attachment. By "x", "y", and "z", you mean the unit vectors in those directions, right? But those are constant vectors (not only constant length, 1, but constant direction) so of course their derivative (and "div" and "curl" are differential operators) is 0.

A simple example of a variable unit vector is . Then and
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January 2nd, 2014, 08:38 PM   #3
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Re: Curl and divergence of units vectos


Now can you see?
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January 5th, 2014, 08:27 AM   #4
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Re: Curl and divergence of units vectos

I cannot access any of your images, Jhenrique. Nevertheless, taking the rectangular co-ordinate system on to be , we get the following implications for the vector field ;

.
In addition, because is a unit vector field, we have . Using this information for all three vector fields in addition to what is shown in your image should help us solve the partial differential equations involved.
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January 5th, 2014, 01:09 PM   #5
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Re: Curl and divergence of units vectos

My images ask what is:


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January 6th, 2014, 12:44 AM   #6
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Re: Curl and divergence of units vectos

Hello, Jhenrique. It appears that is from the spherical co-ordinate system, right? Please confirm these for me in proper detail and also let me know what is. May I also conclude that is the rectangular co-ordinate system on ?
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January 14th, 2014, 01:18 PM   #7
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Re: Curl and divergence of units vectos

I use this convention: [attachment=1:1hxlyy1m]coord.JPG[/attachment:1hxlyy1m]

*
[attachment=0:1hxlyy1m]divergenceandcurl.jpg[/attachment:1hxlyy1m]
Attached Images
File Type: jpg coord.JPG (6.4 KB, 318 views)
File Type: jpg divergenceandcurl.jpg (79.5 KB, 318 views)
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January 16th, 2014, 06:00 AM   #8
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Re: Curl and divergence of units vectos

You're right about the divergences and curls of the unit vector fields in the directions of to all be zero. Use the following identities to complete your chart, where I have used bold face to denote the unit vector in the direction of the given variable:









Of course, you also need the chain rule of partial differentiation;
For a sample computation,

(I see on your chart used instead of and instead of .)
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January 17th, 2014, 06:06 AM   #9
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Re: Curl and divergence of units vectos

You understand, I hope, that the curl and divergence are differential operators? Just as the derivative of any constant is 0 so the curl and derivative of any constant vector is 0. Now, by "units vectors", do you mean vectors of length 1 but with varying direction or vectors pointing in the direction of the coordinate axes, but with varying length?
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January 17th, 2014, 06:33 AM   #10
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Re: Curl and divergence of units vectos

Unit vector is a vector of modulus equal to one but that can have different direction wrt another unit vector of other system of coordinate
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