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- Thread starter Lakshya
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Divergence - A measure of how field lines pass through a closed boundary.

Curl - A measure of how field lines are contained within a certain boundary

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You can see immediately that definition of curl via cross product will not generalize to another dimension (since the cross product is defined only for three dimensions).

These are the definitions:

For divergence:

[tex]

div(\vec{F})=lim_{V \rightarrow 0}\frac{\oint_{\partial V}\vec{F} \hat{n}ds}{V}

[/tex]

The integral in the fractions means the flux of the field through the boundary of a closed region that contains the point x at which you calculate the divergence. So dividing by the volume of the space, then taking the limit, means that the divergence is the density of the flux of the field at any point.

For the curl:

[tex]

curl(\vec{F})=lim_{A \rightarrow 0}\frac{\oint_{\partialA}\vec{F}\vec{dr}}{A}

[/tex]

This limit means the circulation density of the field- how much the field tends to rotate around the point where the curl is calculated. Another meaning would be, how an object will rotate if this field represents a force field.

In three dimensions you can prove those limits will take the familiar form of:

[tex]div(\vec{F})=\partial_{x}F_{1}+\partial_{y}F_{2}+\partial_{z}F_{3}[/tex]

[tex]curl(\vec{F})=(\partial_{y}F_{3}-\partial_{z}F_{2}})\hat{i}+(\partial_{z}F_{1}-\partial_{x}F_{3})\hat{j}+(\partial_{x}F_{2}-\partial_{y}F_{1})\hat{k}[/tex]

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