What is the significance of curl of of a vector field.

[jssamp]

I need help understanding the significance of curl and divergence.

I am nearly at the point where I know how to use Greene's, Stokes and the divergence theorems to convert line, surface, and iterated double and triple integrals. I know how the use the curl and div operators and about irrotational and solenoid fields.

I know that a vector field on a simply connected region with 0 curl, an irrotational field, it is the gradient of a scalar potential function and a line integral is path independent.

I know if the divergence of a field is 0 then it is a solenoid field, not conservative, has sinks/sources, and a stream function can be found.

It's still new to me but I will become comfortable with all of this. But my problem is that is not good enough. I don't want to just know how to use the tools, I want to develop an intuitive understanding of curl and divergence. I have spent a lot of time and effort learning words and equations that are useful for making line integrals easier to deal with. But I feel like I am missing something. They don't usually name theorems after guys for discovering a cool conversion formula.

What is the significance of curl and divergence? I still don't have a good mental picture of them. They are somehow connected to electric and magnetic fields. Brings to mind a uniform E field and a circular B field around a straight thin current. Maxwell's equations include both curl ond div of E and B. The problem with math classes is abstraction. I need applications to really understand it. I transfer to OSU to start pro school in the summer but so far haven't found many at my community college who could help me with this. I checked out the lone book on vector analysis at the library. It helped with the equations but the real understanding seems to elude me. Any help you guys have to offer I will accept with gratitude.

 

[R136a1]

Check out this website and the other pages in there: http://mathinsight.org/curl_idea

 

[Meir Achuz]

Curl is a measure of the rate of change of a vector field in a direction perpendicular to the direction of the vector. As with many mathematical objects, you will get a better understanding of it as you go on using it.

 

[WannabeNewton]

I think that fluid flow is the best setting for the painting of an intuitive picture of curl and divergence. Imagine a fluid with velocity field $\vec{v}$. The curl $\vec{\nabla}\times\vec{v}$ can be interpreted as follows: given a single fluid element, the curl measures the rotation of infinitesimally neighboring fluid elements about the given fluid element. Imagine now a small sphere of fluid elements centered about a given fluid element; the divergence $\vec{\nabla}\cdot\vec{v}$ measures the rate at which this sphere contracts or expands.

 

[JanEnClaesen]

I think that fluid flow is the best setting for the painting of an intuitive picture of curl and divergence. Imagine a fluid with velocity field $\vec{v}$. The curl $\vec{\nabla}\times\vec{v}$ can be interpreted as follows: given a single fluid element, the curl measures the rotation of infinitesimally neighboring fluid elements about the given fluid element. Imagine now a small sphere of fluid elements centered about a given fluid element; the divergence $\vec{\nabla}\cdot\vec{v}$ measures the rate at which this sphere contracts or expands.

Is it related to spherical coordinates?