Understanding the Physical Meaning of Divergence and Curl in Vector Fields

In summary, the divergence of a vector field represents the flux per volume, or flux density, of the field. It is calculated by taking the limit of the flux through a small volume divided by the volume itself. In a fluid, it represents sources or sinks of fluid. The curl, on the other hand, represents the rotation of the fluid and can be thought of as the rate and axis of rotation for a pinwheel placed in the fluid. Both the divergence and curl have important applications in fields such as fluid dynamics and electromagnetism.
  • #1
Nick R
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Hello I am trying to get my head around what the divergence actually represents physically.

If you have some vector field v, and the components of v, vx, vy, vz have dimensions of kg/s ("flow" - mass of material per second) the divergence will have units of kg/(s*m) (mass per time distance)

Say the divergence of v is constant in some region R with volume a.

div(v)*a has units (kg*m^2)/s (mass area per time) - this is the flux of v through area(R)

(div(v)*a)/area(R) has units (mass per time) - the net mass flowing out of R in some time

So what exactly is divergence - kg/(s*m)

Would it be accurate to think of the divergence as "Flux per volume" in general?
 
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  • #2
its where the flux lines end
 
  • #3
Nick R said:
Would it be accurate to think of the divergence as "Flux per volume" in general?

Exactly.

Take some small volume V. Let [itex]\Phi[/itex] be defined as the flux of a vector field u out of the volume V. That is, if S is the closed surface bounding V, then [itex]\Phi[/itex] is the flux across S, in the outward direction. Then the divergence is given by the limit

[tex]\mathrm{div} \vec u = \lim_{V \rightarrow 0} \frac{\Phi}{V}[/tex]
 
  • #4
exactly. Or to be more explicit on the above result,
[tex] \mathrm{div}\vec{u} = \frac{\oint_{S} \vec{u}\cdot \vec{dS}}{V} [/tex].
In fact we can derive this using the mean value theorem on integrals on the divergence theorem. :smile:
 
  • #5
Divergence is simply flux density, flux per volume. In different vector fields the flux density can vary at different points in space. If you add all of the fluxes/volume up and multiply by volume then you will get the total flux through the solid.
 
  • #6
What is the usefulness in knowing the Divergence of a vector field? I mean I realize it is important with regards to stuff like Maxwells equations. But I only learned those in Integral Form, and not differential form.

Also can someone conceptually explain what the Curl represents?
 
  • #7
It's easiest to think of what these operators mean in a fluid.

The divergence in a fluid represents a source or a sink; if there is a point in space where the divergence is nonzero, then at that point, there is fluid being created or destroyed.

The curl represents the rotation of the fluid (imagine eddies of swirling water here). If you could put a little pinwheel in the fluid, the curl gives the rate at which the pinwheel would rotate, and the axis around which that rotation occurs.
 

1. What is a vector field?

A vector field is a mathematical function that assigns a vector to each point in a given space. The vector at a particular point represents the direction and magnitude of the field at that point.

2. What does divergence of a vector field mean?

Divergence of a vector field is a measure of the rate at which the field flows away from or towards a given point. It is calculated by taking the dot product of the gradient operator with the vector field, and it can be either positive (outward flow) or negative (inward flow).

3. How is divergence of a vector field represented mathematically?

The mathematical representation of divergence is ∇ · F, where ∇ is the gradient operator and F is the vector field. It is also sometimes written as div F or div(F).

4. What is the physical significance of divergence?

Divergence has important physical significance in fields such as fluid dynamics and electromagnetism. In fluid dynamics, it represents the rate of change of fluid flow at a given point, while in electromagnetism, it represents the strength of the electric or magnetic field at a point.

5. How is divergence used in real-world applications?

Divergence is used in a variety of real-world applications, such as weather forecasting, fluid flow analysis, and image processing. It is also used in the calculation of flux, which is important in physics and engineering for understanding the flow of energy and particles through a surface.

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