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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
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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 $\vec{v}= x\vec{i}+ \sqrt{1- x^2}\vec{j}$. Then $\nabla\cdot\vec{v}= \frac{\partial x}{\partial x}\vec{i}+ \frac{\partial \sqrt{1- x^2}}{\partial y}\vec{j}$ and $\nabla\times\vec{v}= \frac{-x}{\sqrt{1- x^2}}\vec{k}$

 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 co-ordinate system on $\mathbb{R}^3$ to be $t_1 , t_2 , t_3$, we get the following implications for the vector field ${\bf x}(t_1 , t_2, t_3)$; $\nabla . {\bf x}= 0 \ \Rightarrow \ \dfrac{\partial x_1}{\partial t_1} + \dfrac{\partial x_2}{\partial t_2} + \dfrac{\partial x_3}{\partial t_3} = 0$ $\nabla \times {\bf x}= 0 \ \Rightarrow \ \dfrac{\partial x_3}{\partial t_2} - \dfrac{\partial x_2}{\partial t_3} = \dfrac{\partial x_3}{\partial t_1} - \dfrac{\partial x_1}{\partial t_3} = \dfrac{\partial x_2}{\partial t_1} - \dfrac{\partial x_1}{\partial t_2} = 0$. In addition, because ${\bf x}$ is a unit vector field, we have $x_1^2 + x_2^2 + x_3^2= 1$. 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: $\\ \nabla \cdot \hat{x}=? \\ \nabla \cdot \hat{y}=? \\ \nabla \cdot \hat{z}=? \\ \\ \nabla \cdot \hat{r}=? \\ \nabla \cdot \hat{\theta}=? \\ \nabla \cdot \hat{z}=? \\ \\ \nabla \cdot \hat{\rho}=? \\ \nabla \cdot \hat{\phi}=? \\ \nabla \cdot \hat{\theta}=?$ $\\ \nabla \times \hat{x}=\vec{?} \\ \nabla \times \hat{y}=\vec{?} \\ \nabla \times \hat{z}=\vec{?} \\ \\ \nabla \times \hat{r}=\vec{?} \\ \nabla \times \hat{\theta}=\vec{?} \\ \nabla \times \hat{z}=\vec{?} \\ \\ \nabla \times \hat{\rho}=\vec{?} \\ \nabla \times \hat{\phi}=\vec{?} \\ \nabla \times \hat{\theta}=\vec{?}$
 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 $\theta, \phi , r$ is from the spherical co-ordinate system, right? Please confirm these for me in proper detail and also let me know what $\rho$ is. May I also conclude that $x, y, z$ is the rectangular co-ordinate system on $\mathbb{R}^3$?
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]

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[attachment=0:1hxlyy1m]divergenceandcurl.jpg[/attachment:1hxlyy1m]
Attached Images
 coord.JPG (6.4 KB, 318 views) divergenceandcurl.jpg (79.5 KB, 318 views)

 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 $x, y, z$ 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: ${\bf r}= \cos \theta i + \sin \theta j$ ${\bf \theta}= -\sin \theta i + \cos \theta j$ ${\bf \rho}= \sin \theta \cos \varphi i + \sin \theta \sin \varphi j + \cos \theta k$ ${\bf \varphi}= \cos \theta \cos \varphi i + \cos \theta \sin \varphi j - \sin \theta k$ $x= r\cos \theta = \rho \sin \varphi \cos \theta$ $y= r\sin \theta = \rho \sin \varphi \sin \theta$ $z= \rho \cos \varphi$ $\nabla . v= \frac{\partial v_1}{\partial x} + \frac{\partial v_2}{\partial y} + \frac{\partial v_3}{\partial z}$ $\nabla \times v= \left( \frac{\partial v_3}{\partial y} - \frac{\partial v_2}{\partial z} \right) i - \left( \frac{\partial v_3}{\partial x} - \frac{\partial v_1}{\partial z} \right)j + \left( \frac{\partial v_2}{\partial x} - \frac{\partial v_1}{\partial y} \right) k$ Of course, you also need the chain rule of partial differentiation; $\dfrac{\partial f}{\partial x_i}= \dfrac{\partial f}{\partial t}.\dfrac{\partial t}{\partial x_i}$ For a sample computation, $\nabla . {\bf r}= \dfrac{\partial \cos \theta}{\partial x} + \dfrac{\partial \sin \theta}{\partial y} \\ = \dfrac{\partial \cos \theta}{\partial \theta}.\dfrac{\partial \theta}{\partial x} + \dfrac{\partial \sin \theta}{\partial \theta}.\dfrac{\partial \theta}{\partial y} \\= \frac{-\sin \theta}{-r\sin \theta} + \frac{\cos \theta}{r\cos \theta} = \frac{2}{r}$ (I see on your chart $R$ used instead of $\rho$ and $\phi$ instead of $\varphi$.)
 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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