Understanding the Flow of a Vector Field: Basics and Calculations

Click For Summary
SUMMARY

The discussion centers on the definition and calculation of flow generated by a vector field, specifically in the context of curves and surfaces in R3. The user seeks clarity on the expression of the vector field V(hat) = (v,u), which is identified as potentially misleading since V(hat) typically denotes a unit vector. Participants clarify that the vector field itself represents the flow, and the correct interpretation of V should be V = (v,u) to denote the flow at each point (u,v). The conversation emphasizes the need for precise terminology in vector field analysis.

PREREQUISITES
  • Understanding of vector fields in R3
  • Familiarity with the concept of flow in vector calculus
  • Knowledge of unit vectors and their significance
  • Basic principles of curves and surfaces in three-dimensional space
NEXT STEPS
  • Study the mathematical definition of flow in vector fields
  • Learn about the properties of vector fields in R3
  • Explore the implications of unit vectors in vector calculus
  • Investigate the relationship between curves, surfaces, and vector fields
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are looking to deepen their understanding of vector fields and their applications in three-dimensional space.

ductape
Messages
18
Reaction score
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.
 
Physics news on Phys.org
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:
 
read arnol'd
 

Similar threads

Graduate About ellipticity and a proof that a system of PDEs is elliptic
  • · Replies 0 ·
Replies
0
Views
3K
Undergrad Boundary Conditions For Modelling of a Fluid Using Euler's Equations
  • · Replies 16 ·
Replies
16
Views
3K
Undergrad Is the Poynting Vector Field the Only Factor in Energy Flow in Circuits?
  • · Replies 16 ·
Replies
16
Views
4K
MATLAB MATLAB: Fluid Flow - Curl of a Vector Field
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Constructing left invariant vector fields on SO(3)
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Lie dragging vs Fermi-Walker transport along a given vector field
  • · Replies 26 ·
Replies
26
Views
2K
Undergrad How to define a vector field?
  • · Replies 23 ·
Replies
23
Views
4K
Undergrad Parallel transport vs Lie dragging along a Killing vector field
  • · Replies 20 ·
Replies
20
Views
3K
Undergrad The vector to which a dual vector corresponds
  • · Replies 13 ·
Replies
13
Views
5K
Finding the flow of a vector field
  • · Replies 5 ·
Replies
5
Views
2K