Divergence & Curl: Exploring Meaning & Significance

[Lakshya]

What do Divergence and Curl of a vector function actually mean? They are nice to understand as mathematical operators and then we can work on with them, but what do they mean physically and why are they so important in our study of electromagnetism?

 

[Feldoh]

Applied to E&M:

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

 

[Lakshya]

I have read Vector Calculus from Introduction to Electrodynamics - D.J. Griffiths only. In it, divergence is defined as del dot v and curl as del cross v. And then he tells what do they mean geometrically. How do we arrive at their geometrical meaning from the definition?

 

[elibj123]

It's the other way around - divergence and curl are first defined using a limiting process, so you can generalize it to any dimensions. In the special case of three dimensions, you find that you can represent this operators with the nabla operator.
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:
$$div(\vec{F})=lim_{V \rightarrow 0}\frac{\oint_{\partial V}\vec{F} \hat{n}ds}{V}$$

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:

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

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:

$$div(\vec{F})=\partial_{x}F_{1}+\partial_{y}F_{2}+\partial_{z}F_{3}$$
$$curl(\vec{F})=(\partial_{y}F_{3}-\partial_{z}F_{2}})\hat{i}+(\partial_{z}F_{1}-\partial_{x}F_{3})\hat{j}+(\partial_{x}F_{2}-\partial_{y}F_{1})\hat{k}$$

 

[Lakshya]

Sorry to trouble u guys, I got cleared all my doubts from H.M. Schey's buk Div, Grad, curl and all that. It's a fantastic book!