Deriving vector field line equations from sketches?

[dan38]

Homework Statement

Is it possible to find the vector field line expression without the use of differential equations?
Say I've sketched the field and found the shape to be parabolas, how would I find the general expression by just using the points I've been given?

Homework Equations

The Attempt at a Solution

 

[Simon Bridge]

It is possible - however, it involves being familiar with the form of common vector fields.
Basically you look at the sketch and say "Oh yes - I recognize that one."

So - what do you mean you've "sketched the field and found the shape to be parabolas"?
What, exactly, have you sketched? Vector fields tend to look like a lot of arrows when I sketch them.

 

[dan38]

Basically I've been given a vector equation, V(x,y) and different points to draw the vector field with directional arrows at each point. The question seems to indicate that by sketching the field lines passing through these points I should be able to find the associated differential equation. The shape indicated by following the direction of the tangential vectors is clearly a parabola. So I'm just wondering if it would be possible to find this differential equation...My first thought was to start with the general expression for parabolas (ax^2 + bx + c) but I'm not really sure how to go on from there

 

[Simon Bridge]

Hyperbolas or parabolas?

What do these lines represent?

What does the DE you are thinking of represent?
(What properties of the graph are described by the DE?)

 

[dan38]

Well they look like parabolas becoming wider as I progress down the y-axis...
so dy/dx = ax and y = ax^/2 + c
How do I find the variable "a"?

 

[Simon Bridge]

There are a lot of functions besides parabolas that do that - what reason do you have for assuming they are parabolas?

The approach you are trying won't give you the vector field.
What do those curved lines represent - what is their physical meaning?

 

[dan38]

Well just the shape made it look like parabolas when I joined the dots... I know that the vectors that I have plotted for each point is the curve's tangent vector. And that the curve is defined by the position vector r(t)
Is there something else I'm missing?

 

[Simon Bridge]

The lines should represent the force experienced by a test particle appropriate for the field type.
You need an equation that gets the magnitudes and the directions right.
It's usually easier to use the potentials instead. The question arises: where did the sample vectors come from?

 

[dan38]

Well the sample vectors came from the differential equation
And I already know how I can use that to find the field lines
But the question says to sketch and hence find the field lines, so I thought this way wouldn't be allowed.. :/