Understanding the Flow of a Vector Field: Basics and Calculations

In summary, the conversation discusses the concept of a flow generated by a vector field, specifically looking at curves and surfaces in R3. The speaker is having trouble finding a clear definition and approach for calculating this flow. They mention an example of a vector field with a coordinate expression of (v,u), but are unsure how it relates to the concept of a flow. The conversation also suggests reading Arnold's work for a better understanding of this topic.
  • #1
ductape
18
0
Hello, I am having a lot of trouble finding a definition of a flow generated by a vector field. I can't seem to find a good definition anywhere. I only need a basic definition, and a basic approach to calculating the flow generated by a vector field.
For example, Let U = R2 , x = x(u, v, 0). Let M = x(U). Let V be a vector field who's coordinate expression is as follows: V(hat) = (v,u). What is the flow generated by V. I am only looking at curves and surfaces in R3. Any help would be greatly appreciated.
 
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  • #2
ductape said:
For example, Let U = R2 , x = x(u, v, 0). Let M = x(U). Let V be a vector field who's coordinate expression is as follows: V(hat) = (v,u). What is the flow generated by V. I am only looking at curves and surfaces in R3. Any help would be greatly appreciated.

Hello ductape! :smile:

I don't think there's such a thing as a flow generated by a vector field …

the vector field is the flow …

for example, the vector field (u,v) would be radial.

I don't understand V(hat) = (v,u) either. V(hat) usually means a unit vector, and (v,u) isn't a unit vector. :confused:

If it was V = (v,u), then that would be the flow whose vector is (v,u) at every point (u,v). :smile:
 
  • #3
read arnol'd
 

FAQ: Understanding the Flow of a Vector Field: Basics and Calculations

What is a vector field?

A vector field is a mathematical concept that describes a region of space where each point has a corresponding vector. These vectors have a magnitude and direction, and they represent the physical quantities present in that region.

What is the flow of a vector field?

The flow of a vector field describes the movement of particles or objects that are subject to the influence of the field. It is determined by the direction and magnitude of the vectors at each point in the field.

How is the flow of a vector field calculated?

The flow of a vector field can be calculated using differential equations, where the vectors are described by functions of the spatial coordinates. This allows for the prediction of the behavior of particles in the field over time.

What are some real-life applications of vector fields?

Vector fields are used in various fields of science and engineering, such as fluid dynamics, electromagnetism, and weather forecasting. They can also be used to model the movement of objects in space or the spread of pollutants in the environment.

What is the significance of understanding the flow of a vector field?

Understanding the flow of a vector field is important in many fields of science and engineering as it allows for the prediction and control of the movement of particles or objects. It also provides insights into the underlying physical phenomena and can aid in the development of new technologies.

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