Trying to understand concepts in calculus

[switz5678]

I just finished up a Calculus class which went into fields, and I learned how to find the curl of the field and also the divergence of the field. What I want to know is what does the curl/div of the field actually show us?

Also when you have a field (x,y,z) = (Something)i + (something)j + (something)k does that just model the interaction felt at any given point within that field? Is it representative of a nonuniform field?

I also was wondering how do we derive the equations to represent these complexities in real life scenarios? I'm sure the universe doesn't hand us equations to solve for like class

 

[micromass]

Something to read: http://mathinsight.org/curl_idea

 

[Matterwave]

Something to read: http://mathinsight.org/curl_idea

Hey, that is really cool! I've never seen it explained in such a nice visual way. :D

 

[homeomorphic]

For the divergence, you can imagine a little tiny box. The divergence measure measures the flux out of that box. Actually, it's really the limit of that divided by volume of the box as the volume goes to zero. You can derive the actual expression for the divergence that way.

So, let's just do the flux through the top panel of the box and the bottom. Let's say the field is F(x, y, z).

For the top panel, we just increase z by a little tiny bit to see how strong the field is there, and we get F(x,y,z + dz). But, if you know how flux works, it only measures the normal component of the field that is going through a surface. So, we're really only interested in the z-component of the field, F_z(x,y,dz). The flux through the top will be approximately equal to the area of the top, that is dx dy, times that.

So, the flux through the top panel is F_z(x, y, z+dz) dx dy.

Similarly, for the bottom panel, we get that the flux is -F_z(x, y, z) dx dy.

You may be starting to see where the partial derivative of the z-component might be coming from when you take the limit now.

You can do the flux for all the other panels, divide by the volume, which is dx dy dz, and then you let dx, dy, and dz all go to zero. Then you get the usual expression for the divergence.

You can do the curl in similar way, except it's a little bit messier because the curl is measuring the circulation around the edge of a little square, but that's just for the components of the curl, which is actually a vector field. So, you can do that for a square parallel to each coordinate plane and you get the components of the curl. Then, if you have an oriented square in a different position, the circulation will be given by the dot product of its normal vector with the curl. It takes some thought to work this all out, but that's the general idea. I don't feel like going through all the details here.