Understanding Divergence: Unit Vectors & Magnitude

• salman213
In summary, the conversation discusses the concept of divergence and how it relates to vectors and unit vectors. It is explained that positive divergence occurs when the vector going in is smaller than the vector going out in an infinitesimal region. It is also mentioned that zero divergence means conservation and can be seen in situations where there are no sources or sinks. The example of an electric field with zero divergence except at point charges is given.
salman213
1. I was just trying to understand what divergence means so I hope someone can help me out.

Well from what I have read if I take a vector field and use an infinitesimal region, if the vector going in is smaller than the vector going out there is positive divergence.

Does this mean if i make a circle with UNIT VECTORS there is ZERO divergence. Because what is going in, is the same as going out,

[URL][PLAIN]http://img11.imageshack.us/img11/9028/31455816et8.jpg

If they were NOT unit vectors or vector of the same magnitude then there Would be divergence? is that the correct concept?

The Attempt at a Solution

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Hi salman213!

If you mean unit vectors all going out from a particular point, and a small circle round that point, then there is divergence, because all the vectors are going out of the circle.

Zero divergence means conservation of whatever-it-is …

often you have zero divergence everywhere except at "sources" and "sinks" …

eg an electric field with zero divergence except at the "singularities" where point charges are.

Yes, you are correct. Divergence refers to the rate at which a vector field is expanding or contracting at a given point. If the vectors within an infinitesimal region are getting larger, then there is positive divergence. If they are getting smaller, then there is negative divergence.

In the case of a circle made with unit vectors, the vectors are all the same magnitude and direction, so there is no change in size or direction as you move around the circle. Therefore, there is zero divergence.

However, if the vectors were not unit vectors or if they had different magnitudes, then there would be a change in size or direction as you move around the circle, resulting in some level of divergence.

It's important to note that divergence can also be influenced by the shape of the vector field and the direction of the vectors. So while a circle made with unit vectors may have zero divergence, a different shape or direction of vectors could result in non-zero divergence.

1. What is divergence?

Divergence is a mathematical concept used to measure the flow of a vector field at a specific point. It indicates how much the vector field is expanding or contracting at that point.

2. How is divergence related to unit vectors?

Unit vectors are vectors with a magnitude of 1 that point in a specific direction. Divergence can be calculated by taking the dot product of the vector field with the unit vector in that direction.

3. Why is understanding divergence important?

Understanding divergence is important in many fields of science, including physics, engineering, and fluid dynamics. It allows us to analyze and predict the behavior of vector fields, which can help us understand and solve complex problems.

4. What is the role of magnitude in understanding divergence?

The magnitude of a vector represents its size or length. In the context of divergence, the magnitude of a vector field can indicate the strength or intensity of the flow at a specific point. The magnitude is also used in the calculation of the divergence itself.

5. How can I use unit vectors and magnitude to calculate divergence?

To calculate the divergence of a vector field, you can use the dot product of the field and a unit vector in a specific direction. The magnitude of the resulting vector will give you the divergence at that point. Alternatively, you can use mathematical formulas and techniques to calculate divergence for more complex vector fields.

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