Divergence and Curl of Unit Vectors?

In summary, the conversation discusses the concept of divergence and curl in cartesian and polar coordinates. It is noted that in cartesian coordinates, the divergence and curl of a vector would be zero, but in polar coordinates, there may be non-trivial values. The conversation also touches on the idea of converting theta or phi into something similar to r and z in cylindrical coordinates.
  • #1
cranincu
4
0

Homework Statement


http://img4.imageshack.us/img4/4218/divergenceandcurl.jpg [Broken]

The Attempt at a Solution


Totally confused on what the question's asking. Wouldn't the divergence of say x_hat be the partial of x_hat over x which is just 0? So every answer would just be 0 or something? Same thing goes with the curl? Thanks
 
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  • #2
In cartesian coordinates yes,

But, for example, the r_hat in polar coordinates, written in cartesian coordinates, is
r_hat = (x/sqrt(x^2 +y^2), y/sqrt(x^2 + y^2), 0)

which perhaps has a non-trivial divergence and curl.
 
  • #3
oh i get it

edit: wait, not really. How can I make theta or phi into into something like r and z in cylindrical?
 
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1. What is the concept of divergence and curl?

The divergence and curl are mathematical operations used to describe the behavior of vector fields, which are quantities that have both magnitude and direction. Divergence measures the rate at which a vector field is spreading out or converging at a given point, while curl measures the rotation or circulation of a vector field at a given point.

2. What are the physical applications of divergence and curl?

Divergence and curl have various applications in physics and engineering, particularly in the study of fluid dynamics, electromagnetism, and heat transfer. Divergence is used to model the flow of fluids, while curl is used to describe the behavior of magnetic and electric fields.

3. How are divergence and curl related to unit vectors?

Unit vectors are vectors with a magnitude of 1 and are used to describe the direction of a vector field. The divergence of a unit vector field is always zero, as unit vectors do not have a tendency to spread out or converge. The curl of a unit vector field is also zero, as unit vectors do not rotate or circulate.

4. What is the significance of the divergence theorem in relation to unit vectors?

The divergence theorem, also known as Gauss's theorem, states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of that vector field over the enclosed volume. For a unit vector field, this means that the total number of unit vectors flowing out of a closed surface is equal to the number of unit vectors inside that surface.

5. How can divergence and curl be calculated for a unit vector field?

To calculate the divergence of a unit vector field, take the dot product of the gradient operator and the unit vector field. To calculate the curl of a unit vector field, take the cross product of the curl operator and the unit vector field. These operations can be done using vector calculus techniques, such as partial derivatives and cross products.

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