How Does the Vector Field F = [-x³, x, 0] Behave for Constant x Values?

[uzman1243]

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?

 

[jedishrfu]

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.

 

[uzman1243]

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:

 

[jedishrfu]

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.

 

[uzman1243]

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. :)

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?

 

[jedishrfu]

Compute the magnitude and you'll see it depends only on x.

 

[uzman1243]

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.

 

[jedishrfu]

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.

 

[uzman1243]

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?

 

[jedishrfu]

Didn't you post the wrong image?

 

[uzman1243]

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

 

[jedishrfu]

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)

...

 

[uzman1243]

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?

 

[jedishrfu]

Not that I know of. The only application I can think of is keeping students busy learning new things.