What is the physical significance of the divergence?

In summary, the conversation discusses the concept of divergence in multi-variable calculus and its physical interpretation in terms of a vector field. Divergence is defined as the net movement of field lines away from or towards a point, and can be used to determine the presence of sources or sinks in the field. The concept of curl is also briefly mentioned as a measure of rotation in the field. The conversation highlights the importance of understanding these concepts in fields such as electromagnetics and suggests that they should be taught to students in multi-variable calculus.
  • #1
seang
184
0
Hello;

I remember the days of muti variable calculus. The man said that divergence is equal to del dot the vector field. So on the exam he gave us a vector field, and I did del dot the given vector field and won big time.

The other day I decided my concentration would be electromagnetics. Now I need to know what divergence means. I understand that divergence gives you the scalar value of the source or sink at a point. Right?

It seems weird to me. That you decide the scalar value of a source or a sink at a POINT by considering the WHOLE vector field. I think I need help clearing this up.

For example, let's say I'm given a vector field A. Let's say del dot A = something. Does this mean that the vector field has a source equal to that something? At what point exactly? Is there only one source or sink?

This is the best that I can explain my troubles. I hope someone can help me. Thank you.
 
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  • #2
If you consider a vector field as a physical field (for example, an electric field, or a flow of water), basically the divergence tells you how many field lines move away from the point (net). If the divergence is zero, as many lines originate and terminate at the point (this is for example, the reason that div B = 0 in electrodynamics: if it weren't, magnetic field lines could start somewhere which would prove the existence of magnetic monopoles). When the divergence is positive, more lines start at a point than terminate; basically: the greater the divergence, the more lines will start at the point. In other words: if I view the vector field as describing the flow of water, and I drop a ball near a point, the divergence will tell me if the ball will flow away from or towards the point.

The curl on the other hand (del cross) says something about the rotation: if I drop the same ball in the water, in how far will it flow around the point? The curl will be maximal if the water flows in a circle around my point, it will be zero if it flows radially away from it, for example (in the first case the divergence would be zero, in the latter it would be maximal).

PS Note that divergence and rotation are actually mathematically defined objects, and that this intuitive explanation can help you understand them. But even though they are called divergence and rotation for a reason, your intuition can deceive you.
 
  • #3
that was really helpful, thanks. Between you and a TA friend of mine I think I've got this mostly figured out.

This stuff is really so interesting, when you apply it to something like EM. I can't believe they don't show multi variable students something like this. It would be highly motivational I feel.
 
  • #4

1. What is the physical significance of the divergence?

The divergence is a measure of the net flow of a vector field out of a given point. It tells us whether the vector field is converging or diverging at that point. In physical terms, it represents the rate of outward expansion or compression of a fluid or gas.

2. How is the divergence related to fluid flow?

In fluid dynamics, the divergence of a velocity field indicates the presence of sources (positive divergence) or sinks (negative divergence) of fluid. This is important in understanding the behavior and movement of fluids in various applications, such as in weather patterns or fluid dynamics simulations.

3. Can you give an example of the physical significance of divergence?

An example of the physical significance of divergence is in weather forecasting. By analyzing the divergence of wind patterns, meteorologists can predict areas of low or high pressure, which can indicate the likelihood of storms or calm weather.

4. How is the concept of divergence used in other fields of science?

The concept of divergence is not limited to fluid dynamics and can be applied in other fields such as electromagnetism and thermodynamics. In electromagnetism, the divergence of the electric and magnetic fields is used to study the behavior of charged particles and electromagnetic waves. In thermodynamics, the divergence of a heat flow field is used to analyze heat transfer and energy conservation.

5. Why is the divergence important in vector calculus?

The divergence is an important concept in vector calculus because it allows us to mathematically describe and understand the behavior of vector fields. It is a fundamental concept in the study of fluid dynamics, electromagnetism, and other fields where vector fields are present. Additionally, it plays a crucial role in many physical laws and equations, such as the continuity equation and Gauss's law.

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