Gradient, divegrance and curl? del operator

In summary: S .. thnx!In summary, the gradient, divergence, and curl are all descriptions of vector fields in terms of their rate of change with respect to space. The dot and cross products are still physically undefined clearly to me.
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Lonley_Shepherd
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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!
 
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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
 
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  • #4
Lonley_Shepherd said:
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
 
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1. What is the gradient?

The gradient is a mathematical operator, denoted by the symbol ∇ (del), that represents the rate of change of a scalar field in three-dimensional space. It is a vector that points in the direction of the greatest increase of a function at a given point.

2. What is divergence?

Divergence is a measure of the flow of a vector field out of a given point. It is represented by the dot product of the del operator (∇) and the vector field. A positive divergence indicates that the vector field is spreading out from that point, whereas a negative divergence indicates that the vector field is converging towards that point.

3. What is curl?

Curl is a measure of the rotation of a vector field around a given point. It is represented by the cross product of the del operator (∇) and the vector field. A positive curl indicates that the vector field is rotating in a counterclockwise direction, whereas a negative curl indicates a clockwise rotation.

4. How are gradient, divergence, and curl related?

Gradient, divergence, and curl are all operations performed on vector fields using the del operator (∇). Gradient represents the rate of change of a scalar field, while divergence and curl represent the flow and rotation of a vector field, respectively. These operations are important in the study of fluid mechanics, electromagnetism, and other areas of physics.

5. What is the physical significance of the del operator?

The del operator (∇) is a mathematical tool used to describe the physical properties of vector fields. It allows us to calculate fundamental quantities such as gradient, divergence, and curl, and is essential in solving many problems in physics and engineering. The del operator also has geometric significance, as it represents the direction and magnitude of change in a vector field at a given point.

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