Exploring the Behavior of a Vector Field with Constant x Values

In summary, the vector field F = [ -x3 , x, 0] behaves differently for constant x values, with the vector field in the y-z plane having a constant vector value and changing only with x. It is not generally parallel to the x unit vector. As the value of x increases, the vector field remains the same in the y-z plane. There is no z component in the vector field, so the vectors will always reside in the xy plane for any z value. The magnitude of the vector field depends solely on the value of x. There is no known natural application for this vector field.
  • #1
uzman1243
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vector field F = [ -x3 , x, 0]

how does this behave for a constant x? Does that mean that I plot the vector field for different values of x?
 
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  • #2
When x is constant the vector field in that plane of y and z has the same constant vector value. It changes only with x and it isn't in general parallel to the x unit vector.

It's like the ideal notion of gravity on the Earth's surface where it changes only with height but not with moving east or west or north or south.
 
  • #3
jedishrfu said:
When x is constant the vector field in that plane of y and z has the same constant vector value. It changes only with x and it isn't in general parallel to the x unit vector.

ah yes. I see. But what happens to the vector field orientation as values of x increases? would you get something like this:
Lambda-Cold_Dark_Matter,_Accelerated_Expansion_of_the_Universe,_Big_Bang-Inflation.jpg
 
  • #4
I guess I don't understand your question. We went from vector field to the big bang in one post.

If you look at the vector field you have, you can immediately see that there is no z component meaning the vector field vectors will all reside in the xy plane for any z value. Also since the xy components depend on x alone then the vectors for a given y and z coordinate will be the same ie point in the same direction and have the same magnitude.

Why not try drawing the vector field at each x,y,z coordinate and see how it will look.
 
  • #5
jedishrfu said:
I guess I don't understand your question. We went from vector field to the big bang in one post.
Oops I posted the wrong image. Long night, sorry. :)

jedishrfu said:
If you look at the vector field you have, you can immediately see that there is no z component meaning the vector field vectors will all reside in the xy plane for any z value. Also since the xy components depend on x alone then the vectors for a given y and z coordinate will be the same ie point in the same direction and have the same magnitude.

How can it have the same magnitude?
 
  • #6
Compute the magnitude and you'll see it depends only on x.
 
  • #7
jedishrfu said:
Compute the magnitude and you'll see it depends only on x.
Yes. Thank you so much.

Just one more question, as the value of x increases, how does the vector field look like? I can't find a way to plot this online to visualize it. I want to see this vector field in order to think of a scenario in nature where this vector field applies to.
 
  • #8
Just plot it in 2d that is in x and y. It will be the same for every value of z so that means it's like stacking the x y planes on top of one another.
 
  • #9
jedishrfu said:
Just plot it in 2d that is in x and y. It will be the same for every value of z so that means it's like stacking the x y planes on top of one another.
So the planes stacked together will form something like the image I posted above?
 
  • #11
jedishrfu said:
Didn't you post the wrong image?
Oh yes. Would it be field lines given by the equation y = 1/x +c stacked up on z axis? I got the equation by differentiating the vector field f = dr/dt
 
  • #12
No, I don't think you need to differentiate anything.

pick a point say (0,0,0) and the vector at that point is (0,0,0)

Pick another point (2,0,0) and the vector at that point is (-8,2,0)

...
 
  • #13
jedishrfu said:
No, I don't think you need to differentiate anything.

pick a point say (0,0,0) and the vector at that point is (0,0,0)

Pick another point (2,0,0) and the vector at that point is (-8,2,0)

...
I see. Is there any application of this in real life? As in, is there any part of nature that uses this vector field that you can think of?
 
  • #14
Not that I know of. The only application I can think of is keeping students busy learning new things.
 
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1. What is a vector field?

A vector field is a mathematical concept that describes the behavior of vectors at different points in space. It can be represented as a set of arrows, each pointing in a specific direction and magnitude, to visualize the direction and strength of a vector at a particular point.

2. How is a vector field useful?

Vector fields are useful in many scientific fields, such as physics, engineering, and computer graphics. They can describe the flow of fluids, the movement of particles, and the direction of forces in a given space, helping to understand and predict complex systems.

3. How do you create a vector field?

A vector field is typically created by specifying a mathematical function that defines the direction and magnitude of vectors at different points in space. This can be done using software programs or by hand using equations and graphs.

4. What is the difference between a scalar field and a vector field?

A scalar field has a single value (scalar) at each point in space, whereas a vector field has a vector (magnitude and direction) at each point. Scalar fields are used to describe properties such as temperature or pressure, while vector fields describe the direction and strength of a quantity.

5. How can vector fields be applied in real-world situations?

Vector fields have many practical applications, such as in weather forecasting, fluid dynamics, and computer graphics. They can be used to predict the movement of air or water currents, simulate the behavior of particles in a fluid, and create realistic visual effects in movies and video games.

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