Gradient, divegrance and curl? del operator

[Lonley_Shepherd]

Gradient, divegrance and curl? del operator!

in static magnetic and electric fields, the del operator was introduced and then used to describe three different quantities.. i still can't quite figure out the physical meaning and difference between the curl,divergance and the gradient in terms of vector fields.. also the dot and cross products are still physically undefined clearly to me :S .. thnx!

 

[Astronuc]

See this discussion on the gradient
https://www.physicsforums.com/showthread.php?t=86874

this for the divergence
http://hyperphysics.phy-astr.gsu.edu/hbase/diverg.html#c1

and this for the curl
http://hyperphysics.phy-astr.gsu.edu/hbase/curl.html#c1

And vector calculus -
http://hyperphysics.phy-astr.gsu.edu/hbase/vecal.html#c1
http://hyperphysics.phy-astr.gsu.edu/hbase/vecal2.html#c1

And I need to finish what I promised in the other thread.

 

[Lonley_Shepherd]

ohh please do finish wut u promised! and well the links to that other site are just stating what div and curl are... i do get wut they mean mathematically... del dot and del cross... but wen related to electric and magnetic fields.. it just isn't clear anymore.. how can a rate of change with respect to space be a scalar quantity.. and by the cross product of del to a vector the rate of change with respect to space has a direction! just like the gradient then..
i read some texts abt the curl that refer to it as the net rotation of a field... how kan a uniform magnetic field rotate.. and when i say rotate i surely must specify an axis of rotation.,. aaaaaaaaaaaa ,.. maybe am smart enough to breath only!

PS: i dunt think it was right moving my discussion here.. its physics wuts bothering me.. but that's ur call :D

 

[nrqed]

in static magnetic and electric fields, the del operator was introduced and then used to describe three different quantities.. i still can't quite figure out the physical meaning and difference between the curl,divergance and the gradient in terms of vector fields.. also the dot and cross products are still physically undefined clearly to me :S .. thnx!

Very roughly...(I will assume you know what a scalar field and a vector field are.. A scalar field assigns a number (i.e. a scalar) to all points in space and a vector field assigns a vector to all points in space)

The gradient of a scalar roughly tells at each point the direction in which the scalar field increases the most and how large the change is at that point is (think of a scalar field as a surface, for example the surface of mountains and valleys. The gradient at a given point shows in what direction the surface goes up the steepest and how steep the surface is at that point. If part of the surface is flat, the gradient there is zero).

For the divergence of a vector field, consider a tiny (infinitesimal) volume (a cube, say). The divergence of the field at the center of the cube times the volume of the cube is the net flow of the vector field through the sides of the cube. If the divergence is positive for example, there is a net flow of the vector field out of the cube.

For the curl, consider a tiny loop (in a circle let's say). The curl of a vector field at the center of the loop is equal to the "circulation" of the vector field along the loop. In the image of the water velocity field, a nonzero curl at a point indicates that there is a net rotation of water around that point (like a tiny vortex).

This is all very crude but I just meant to give a very brief physical interpretation.

Hope this helps


Patrick