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January 1st, 2014, 04:12 PM  #1 
Senior Member Joined: Nov 2013 Posts: 137 Thanks: 1  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 
January 2nd, 2014, 06:27 AM  #2  
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Curl and divergence of units vectos Quote:
A simple example of a variable unit vector is . Then and  
January 2nd, 2014, 08:38 PM  #3 
Senior Member Joined: Nov 2013 Posts: 137 Thanks: 1  Re: Curl and divergence of units vectos Now can you see? 
January 5th, 2014, 08:27 AM  #4 
Senior Member Joined: Dec 2012 Posts: 372 Thanks: 2  Re: Curl and divergence of units vectos
I cannot access any of your images, Jhenrique. Nevertheless, taking the rectangular coordinate 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. 
January 5th, 2014, 01:09 PM  #5 
Senior Member Joined: Nov 2013 Posts: 137 Thanks: 1  Re: Curl and divergence of units vectos
My images ask what is: 
January 6th, 2014, 12:44 AM  #6 
Senior Member Joined: Dec 2012 Posts: 372 Thanks: 2  Re: Curl and divergence of units vectos
Hello, Jhenrique. It appears that is from the spherical coordinate 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 coordinate system on ?

January 14th, 2014, 01:18 PM  #7 
Senior Member Joined: Nov 2013 Posts: 137 Thanks: 1  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] 
January 16th, 2014, 06:00 AM  #8 
Senior Member Joined: Dec 2012 Posts: 372 Thanks: 2  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 .) 
January 17th, 2014, 06:06 AM  #9 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  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?

January 17th, 2014, 06:33 AM  #10 
Senior Member Joined: Nov 2013 Posts: 137 Thanks: 1  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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