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i am having trouble with understanding the physical significance of these two operators.

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i am having trouble with understanding the physical significance of these two operators.

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Pythagorean

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i am having trouble with understanding the physical significance of these two operators.

they're somewhat like they sound. Divergence represents change as you get away from a point. Central forces like gravity are divergent: as you get farther away from a source point, the effects change, no matter what direction you go, as long as it's away, in the r-hat direction.

Curl is more synonymous with the magnetic field or turbulent flow. It describes more how things change as you circumnavigate the point.

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mathwonk

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to measure curl, in the plane, put a little circle centered at the point, and dot the vector field with the tangential vector of the circle, i.e. measure the tendency of the field to rotate around the circle, and integrate, and divide by the area of the circle, then let the radius go to zero.

this seems intuitively pretty much like it anyway.

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The interactive models allow you to really get a feel for what mathwonk was saying.

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CompuChip

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And also see this thread on PF.

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The physics interpretation of divergence and curl,it is possible to be seen very fast going to an infinitesimal cube.

In Cartesian coordinates, supposing an infinitesimal cube, trim in the origin, we can serve to us as the differential of the field to see its meaning:

[tex] \ \vec{\nabla} \cdot \vec{v} =\underbrace{ \frac{1}{\tau} \underbrace{\oint_S \vec{v} \cdot \vec{ds}}_{\text{flow of field through S}}}_{\text{flow per unit volume of field through S}} [/tex]

Considering that the flow is the coordinate of v that is perpendicular to each face of the cube multiplied by their area, we have:

[tex]- v_x dxdz + \left( \underbrace{v_x + \frac{\partial v_x}{\partial x} dx}_{\text{ infinitesimal increase of the field on x-axis}} \right) dxdz [/tex]

If you operate this with all the faces of the cube, you will see that you obtain the divergence.

So, we can conclude, that the physical meaning of divergence, is the flow of the field by volume unit.

For the curl, the reasoning is analogous

In Cartesian coordinates, supposing an infinitesimal cube, trim in the origin, we can serve to us as the differential of the field to see its meaning:

[tex] \ \vec{\nabla} \cdot \vec{v} =\underbrace{ \frac{1}{\tau} \underbrace{\oint_S \vec{v} \cdot \vec{ds}}_{\text{flow of field through S}}}_{\text{flow per unit volume of field through S}} [/tex]

Considering that the flow is the coordinate of v that is perpendicular to each face of the cube multiplied by their area, we have:

[tex]- v_x dxdz + \left( \underbrace{v_x + \frac{\partial v_x}{\partial x} dx}_{\text{ infinitesimal increase of the field on x-axis}} \right) dxdz [/tex]

If you operate this with all the faces of the cube, you will see that you obtain the divergence.

So, we can conclude, that the physical meaning of divergence, is the flow of the field by volume unit.

For the curl, the reasoning is analogous

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