Determining Field Lines of a Vector Field in R^3

In summary, the problem asks for the determination of field lines for a vector field, f: R^3 --> R^3, defined by f(x,y,z) = (x/(1+z^2), y/(x^2+y^2), 0). The solution involves finding the vector field, F(x,y,z) = (x/((1+z^2)*(x^2+y^2))*i + (y/((1+z^2)*(x^2+y^2))*j + 0*k, and using a chosen point, (1,1,1), to find the coordinates for a 3D graph. However, there are difficulties in plotting the coordinates and determining the z-coordinate.
  • #1
Kork
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Homework Statement



Determine the field lines of the vector field f: R^3 --> R^3 defined by

f(x,y,z) = (x/(1+z2) , y/(x2+y2) , 0)

Homework Equations


The Attempt at a Solution



I know that my vector field is:

F(x,y,z) = (x/((1+z^2)*(x^2+y^2))*i + (y/((1+z^2)*(x^2+y^2))*j + 0k

I choose the point (1,1,1) and if I put that into F(1,1,1) I get

(1/8)i + (1/8)j + 0*k --> <1/8, 1/8>

I need a 3D graph right?

Well, first of all, I don't know how to plot (1/8,1/8) into a 3D graph. Second of all, what happens to the z-coordinate?
Third: I want more than one vector obviously, but when I try with (1,1,-1) I get ∞, which I cannot use to make a vector. Which coordinates can I use?
 
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  • #2
i would like to know this as well
 

Related to Determining Field Lines of a Vector Field in R^3

1. What is the purpose of determining field lines?

The purpose of determining field lines is to visualize and understand the direction and strength of a force field in a given space. This is important in many scientific fields, such as physics, engineering, and geology.

2. How do you determine the direction of field lines?

The direction of field lines is determined by placing a test charge in the field and observing the direction in which it moves. The field lines will point away from positive charges and towards negative charges.

3. What factors affect the shape of field lines?

The shape of field lines is affected by the strength and distribution of the source of the field, as well as the presence of other charges or objects in the field.

4. Can field lines intersect?

No, field lines cannot intersect. They must either be parallel or form closed loops. If they were to intersect, it would indicate contradictory directions for the force at that point.

5. How do you represent field lines in a diagram?

Field lines are represented in a diagram by drawing lines with arrows to show the direction of the field. The density of the lines can also indicate the strength of the field, with closer lines representing a stronger field.

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